Stephen, how's your testing of removing the advertising of the IMAP IDLE feature going?
I have noted that adjusting my Pihole filtering to stop blocking iCloud Private Relay servers helped substantially - but not completely. Re-enabling the blocking of the servers brings back all the problems. During testing of this I noted that there is likely some interplay with DNS TTL values, which can temporarily make it look like things are fixed when in reality they are not.
I have an open case about this with Apple support; I spoke to my contact person again today and he confirmed that the engineers are aware of the problem and are actively investigating. I've signposted him (again) to this thread. He was keen to point out that they are keen to hear of any fixes, workarounds, etc in support of trying to find ways to reproduce the problem etc. I specifically noted to him the observations made about IDLE and he said he would pass this on.
Fingers remaining crossed for a fix soon.
[Edited by Moderator]