macBook-Air mail stays offline on ms-Exchange account

The mail app works ok with iCloud account but not with the Exchange account. I've added (through system preference | Internet accounts) my Exchange account without problems (the passwd was verified during creation, also later). After adding this account to the mail-app, the account status (in the mail preferences | Accounts | Account information) stays offline. The connection doctor reports "Could not connect to this Exchange account"; its log activity reports: "(on port:587): 235 2.7.0 Authentication successful (CKUSER)". However, in: ... | Accounts | Server settings any Passwd (correct or incorrect) is treated the same way: after clicking "Save" the "Verifying Server Setting" window is displayed for about 2' without any visible effect.

I access the same Exchange-mail account from mail-app on iMac and from older macBook without problems. Therefore, I suspect that the problem is specific to this particular macBook. Maybe it is caused by the dual (windows and macOS) boot installed on it previously. I've removed the window partition but maybe some left-overs are hidden somewhere; maybe some settings need to be changed?. BTW: the ms-oneDrive runs OK.

Summing-up, the question is simple: how to make the Exchange mail account on-line? Please, comment.

The macBook basic data: Air, 13-inch, early 2014, it runs macOS 11.6.

Best wishes,

Marek Makowski