Server is  dovecot 2.1.12-r1 on gentoo running on a private server (I can offer a test account if needed); after some time idling, the server disconnects the client; here follows the server log.

Feb 18 13:01:51 anthrax dovecot: imap-login: Login: user=<jacopods>, method=PLAIN, rip=151.24.23.51, lip=192.168.1.137, mpid=6830, TLS, session=<n/aPff7VnACXGBcz>
Feb 18 13:39:16 anthrax dovecot: imap-login: Login: user=<jacopods>, method=PLAIN, rip=151.24.23.51, lip=192.168.1.137, mpid=6914, TLS, session=<rBZaA//V/wCXGBcz>
Feb 18 14:09:26 anthrax dovecot: imap(jacopods): Disconnected for inactivity in=215 out=771293
Feb 18 14:09:26 anthrax dovecot: imap-login: Disconnected (no auth attempts in 0 secs): user=<>, rip=151.24.23.51, lip=192.168.1.137, TLS, session=<3upCb//VQQCXGBcz>
Feb 18 14:09:56 anthrax dovecot: imap-login: Disconnected (no auth attempts in 30 secs): user=<>, rip=151.24.23.51, lip=192.168.1.137, TLS handshaking: Disconnected, session=<smoLcf/VQgCXGBcz>
Feb 18 14:09:56 anthrax dovecot: imap-login: Disconnected (no auth attempts in 0 secs): user=<>, rip=151.24.23.51, lip=192.168.1.137, TLS handshaking: Disconnected, session=<OrULcf/VQwCXGBcz>
Feb 18 14:09:56 anthrax dovecot: imap-login: Disconnected (no auth attempts in 0 secs): user=<>, rip=151.24.23.51, lip=192.168.1.137, TLS handshaking: Disconnected, session=<aPALcf/VRACXGBcz>
Feb 18 14:10:26 anthrax dovecot: imap-login: Disconnected (no auth attempts in 30 secs): user=<>, rip=151.24.23.51, lip=192.168.1.137, TLS handshaking: Disconnected, session=<nw7Vcv/VRQCXGBcz>
Feb 18 14:10:26 anthrax dovecot: imap-login: Disconnected (no auth attempts in 0 secs): user=<>, rip=151.24.23.51, lip=192.168.1.137, TLS handshaking: Disconnected, session=<JUzVcv/VRwCXGBcz>
Feb 18 14:10:56 anthrax dovecot: imap-login: Disconnected (no auth attempts in 30 secs): user=<>, rip=151.24.23.51, lip=192.168.1.137, TLS handshaking: Disconnected, session=<U2SedP/VSACXGBcz>

After the disconnection, akonadi apparently does not realize it has been disconnected:

C: A000003 CAPABILITY
S: * CAPABILITY IMAP4rev1 LITERAL+ SASL-IR LOGIN-REFERRALS ID ENABLE IDLE SORT SORT=DISPLAY THREAD=REFERENCES THREAD=REFS THREAD=ORDEREDSUBJECT MULTIAPPEND UNSELECT CHILDREN NAMESPACE UIDPLUS LIST-EXTENDED I18NLEVEL=1 CONDSTORE QRESYNC ESEARCH ESORT SEARCHRES WITHIN CONTEXT=SEARCH LIST-STATUS SPECIAL-USE
S: A000003 OK Capability completed.
C: A000004 NAMESPACE
S: * NAMESPACE ( ("" ".") ) ( ) ( )
S: A000004 OK Namespace completed.
C: A000005 LIST "" *
S: * LIST ( \HasNoChildren ) . fail2ban
S: * LIST ( \HasNoChildren ) . Outgoing security alerts
S: * LIST ( \HasNoChildren \Drafts ) . Drafts
S: * LIST ( \HasNoChildren ) . DenyHosts
S: * LIST ( \HasNoChildren \Sent ) . Sent Messages
S: * LIST ( \HasNoChildren ) . Deleted Messages
S: * LIST ( \HasNoChildren ) . INBOX
S: A000005 OK List completed.
C: A000006 SELECT "INBOX"
S: * FLAGS ( \Answered \Flagged \Deleted \Seen \Draft $REPLIED $ATTACHMENT $NOTJUNK )
S: * OK Flags permitted. [ PERMANENTFLAGS ( \Answered \Flagged \Deleted \Seen \Draft $REPLIED $ATTACHMENT $NOTJUNK \* )  ]
S: * 1097 EXISTS
S: * 0 RECENT
S: * OK First unseen. [ UNSEEN 1097  ]
S: * OK UIDs valid [ UIDVALIDITY 1360495157  ]
S: * OK Predicted next UID [ UIDNEXT 1110  ]
S: * OK No permanent modsequences [ NOMODSEQ  ]
S: A000006 OK Select completed. [ READ-WRITE  ]
C: A000007 IDLE
S: + idling
S: * OK Still here

and keeps saying OK Still here even after the disconnection

Reproducible: Always