IIRC, Jan says it isn't.

Fixed. Thank you!

By the way this is not so rare setup. It's standard Cyrus with 
standard global and mailbox settings. No idea why this was not 
reported earler though.

I think this is related to your rare setup where the CONDSTORE 
extension is available, but it is disabled per mailbox.  Without mod 
sequences, we cannot do search caching since flag changes (i.e. 
marking message as seen) will not invalidate the cache ID.  So we need 
to disable search caching for these mailboxes.

Changes have been made in Git for this ticket:

Disable search caching if a mailbox reports it does not support mod 
sequences (Bug #9833).

  3 files changed, 16 insertions(+), 1 deletions(-)
http://git.horde.org/horde-git/-/commit/56be75f9196778b399db7a27f5e91fcb112c03d4

Here you go. Logs were generated the following way:

1. Refresh INBOX with one unread message.
2. Open that message.
3. Go back to INBOX.

Don't see this.  Even more, IMAP logs clearly show that the unseen 
count is being polled every time the mailbox is listed:

(1305268859.4409) C: 5 SEARCH RETURN (COUNT) CHARSET UTF-8 UNDELETED UNSEEN
(1305268859.4788) S: * ESEARCH (TAG "5") COUNT 13

So you will need to provide IMAP debug logs to prove otherwise or else 
I am going to close this ticket.

BTW, deleting any message from the mailbox resets this counter to 
correct number.

Yes, in standard view. I see this under all browsers. Attaching a 
screenshot. Any ideas what related settings I could be running 
different from default?

I'm assuming you mean the standard view?  I don't see this.

Unread message count is not updated in HTML title and Folder title 
above action bar when messages are marked as read. It always just 
increases on new mail. Never decreases.