FWIW, the connection doctor tells me that the SMTP connections are OK but the IMAP ones have the connection issue. The iCloud and Gmail accounts both have the IMAP issue while SMTP is OK. Here is a sample of a log:
INITIATING CONNECTION Sep 14 18:55:32.881 host:smtp.gmail.com -- port:25 -- socket:0x0 -- thread:0x7f9371fa4440
INITIATING CONNECTION Sep 14 18:55:32.881 host:smtp.gmail.com -- port:25 -- socket:0x0 -- thread:0x7f9371fc5ac0
INITIATING CONNECTION Sep 14 18:55:32.882 host:p01-smtp.mail.me.com -- port:587 -- socket:0x0 -- thread:0x7f9374a66e80
INITIATING CONNECTION Sep 14 18:55:32.882 host:mail.licenselab.com -- port:587 -- socket:0x0 -- thread:0x7f93743d2010
CONNECTED Sep 14 18:55:32.909 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
CONNECTED Sep 14 18:55:32.910 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
CONNECTED Sep 14 18:55:32.924 [kCFStreamSocketSecurityLevelNone] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
READ Sep 14 18:55:32.936 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
220 smtp.gmail.com ESMTP m206sm4844774ita.13 - gsmtp
READ Sep 14 18:55:32.936 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
220 smtp.gmail.com ESMTP 194sm563241itu.4 - gsmtp
CONNECTED Sep 14 18:55:32.940 [kCFStreamSocketSecurityLevelNone] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
WROTE Sep 14 18:55:32.967 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
EHLO [10.0.1.43]
WROTE Sep 14 18:55:32.967 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
EHLO [10.0.1.43]
READ Sep 14 18:55:32.987 [kCFStreamSocketSecurityLevelNone] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
220 st11p01im-asmtp002.mac.com -- Server ESMTP (Oracle Communications Messaging Server 7.0.5.38.0 64bit (built Feb 26 2016))
READ Sep 14 18:55:32.991 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
250-smtp.gmail.com at your service, [72.128.122.145]
250-SIZE 35882577
250-8BITMIME
250-STARTTLS
250-ENHANCEDSTATUSCODES
250-PIPELINING
250-CHUNKING
250 SMTPUTF8
READ Sep 14 18:55:32.991 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
250-smtp.gmail.com at your service, [72.128.122.145]
250-SIZE 35882577
250-8BITMIME
250-STARTTLS
250-ENHANCEDSTATUSCODES
250-PIPELINING
250-CHUNKING
250 SMTPUTF8
WROTE Sep 14 18:55:32.992 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
STARTTLS
WROTE Sep 14 18:55:32.992 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
STARTTLS
WROTE Sep 14 18:55:32.996 [kCFStreamSocketSecurityLevelNone] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
EHLO [10.0.1.43]
READ Sep 14 18:55:33.015 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
220 2.0.0 Ready to start TLS
READ Sep 14 18:55:33.015 [kCFStreamSocketSecurityLevelNone] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
220 2.0.0 Ready to start TLS
WROTE Sep 14 18:55:33.038 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
EHLO [10.0.1.43]
WROTE Sep 14 18:55:33.038 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
EHLO [10.0.1.43]
READ Sep 14 18:55:33.046 [kCFStreamSocketSecurityLevelNone] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
250-st11p01im-asmtp002.mac.com
250-8BITMIME
250-PIPELINING
250-CHUNKING
250-DSN
250-ENHANCEDSTATUSCODES
250-EXPN
250-HELP
250-XADR
250-XSTA
250-XCIR
250-XGEN
250-XLOOP 8B9D21047820C8BEE822712C9EFB74DE
250-STARTTLS
250-NO-SOLICITING
250 SIZE 28311552
WROTE Sep 14 18:55:33.047 [kCFStreamSocketSecurityLevelNone] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
STARTTLS
READ Sep 14 18:55:33.062 [kCFStreamSocketSecurityLevelNone] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
220 n28.mail01.mtsvc.net ESMTP Exim 4.84_2 Wed, 14 Sep 2016 19:55:33 -0400
WROTE Sep 14 18:55:33.072 [kCFStreamSocketSecurityLevelNone] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
EHLO [10.0.1.43]
READ Sep 14 18:55:33.097 [kCFStreamSocketSecurityLevelNone] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
220 2.5.0 Go ahead with TLS negotiation.
READ Sep 14 18:55:33.108 [kCFStreamSocketSecurityLevelNone] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
250-n28.mail01.mtsvc.net Hello cpe-72-128-122-145.wi.res.rr.com [72.128.122.145]
250-SIZE 26214400
250-8BITMIME
250-PIPELINING
250-AUTH LOGIN PLAIN
250-STARTTLS
250 HELP
WROTE Sep 14 18:55:33.109 [kCFStreamSocketSecurityLevelNone] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
STARTTLS
READ Sep 14 18:55:33.120 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
250-smtp.gmail.com at your service, [72.128.122.145]
250-SIZE 35882577
250-8BITMIME
250-AUTH LOGIN PLAIN XOAUTH2 PLAIN-CLIENTTOKEN OAUTHBEARER XOAUTH
250-ENHANCEDSTATUSCODES
250-PIPELINING
250-CHUNKING
250 SMTPUTF8
WROTE Sep 14 18:55:33.121 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
AUTH XOAUTH2 dXNlcj1teXN0ZXJ5cm9vbW1hc3RlcmluZ0BnbWFpbC5jb20BYXV0aD1CZWFyZXIgeWEyOS5DamRlQTd nSExTbUFsVEdhUGZnbjRFWXQwT01pdlJzVGZtcDFZWFM1YVU5Wl9JYWQ2T3hEcWhtSVVQczZwUFBOZjN VelZkc1g0cFFkAQE=
READ Sep 14 18:55:33.122 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
250-smtp.gmail.com at your service, [72.128.122.145]
250-SIZE 35882577
250-8BITMIME
250-AUTH LOGIN PLAIN XOAUTH2 PLAIN-CLIENTTOKEN OAUTHBEARER XOAUTH
250-ENHANCEDSTATUSCODES
250-PIPELINING
250-CHUNKING
250 SMTPUTF8
WROTE Sep 14 18:55:33.122 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
AUTH XOAUTH2 dXNlcj1qdXN0aW5jYXJscGVya2luc0BnbWFpbC5jb20BYXV0aD1CZWFyZXIgeWEyOS5DamxlQTNHUVg yWE5OVWJVLTliRGpXcjRlbmVOdHdPQUFTTFRUWEthRFZ5RE5YZS00M1k0SzZQN1dKMklUZ2pKYU05THZ tQ2JVNWVpU1BrAQE=
READ Sep 14 18:55:33.160 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
235 2.7.0 Accepted
WROTE Sep 14 18:55:33.160 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:smtp.gmail.com -- port:25 -- socket:0x7f9374751ec0 -- thread:0x7f9371fc5ac0
QUIT
READ Sep 14 18:55:33.168 [kCFStreamSocketSecurityLevelNone] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
220 TLS go ahead
READ Sep 14 18:55:33.197 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
235 2.7.0 Accepted
WROTE Sep 14 18:55:33.197 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:smtp.gmail.com -- port:25 -- socket:0x7f93745067f0 -- thread:0x7f9371fa4440
QUIT
WROTE Sep 14 18:55:33.300 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
EHLO [10.0.1.43]
READ Sep 14 18:55:33.350 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
250-st11p01im-asmtp002.mac.com
250-8BITMIME
250-PIPELINING
250-CHUNKING
250-DSN
250-ENHANCEDSTATUSCODES
250-EXPN
250-HELP
250-XADR
250-XSTA
250-XCIR
250-XGEN
250-XLOOP 8B9D21047820C8BEE822712C9EFB74DE
250-AUTH PLAIN LOGIN ATOKEN
250-AUTH=LOGIN PLAIN
250-NO-SOLICITING
250 SIZE 28311552
WROTE Sep 14 18:55:33.351 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
AUTH ATOKEN (*** 80 bytes hidden ***)
WROTE Sep 14 18:55:33.391 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
EHLO [10.0.1.43]
READ Sep 14 18:55:33.429 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
250-n28.mail01.mtsvc.net Hello cpe-72-128-122-145.wi.res.rr.com [72.128.122.145]
250-SIZE 26214400
250-8BITMIME
250-PIPELINING
250-AUTH LOGIN PLAIN
250 HELP
WROTE Sep 14 18:55:33.430 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
AUTH PLAIN (*** 76 bytes hidden ***)
READ Sep 14 18:55:33.472 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
235 Authentication succeeded
WROTE Sep 14 18:55:33.472 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:mail.licenselab.com -- port:587 -- socket:0x7f9374219de0 -- thread:0x7f93743d2010
QUIT
READ Sep 14 18:55:33.572 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
235 2.7.0 ATOKEN authentication successful.
WROTE Sep 14 18:55:33.572 [kCFStreamSocketSecurityLevelTLSv1_0] -- host:p01-smtp.mail.me.com -- port:587 -- socket:0x7f93742d53c0 -- thread:0x7f9374a66e80
QUIT