You are correct that Horde can do nothing about a client that doesn't 
want to push. "Push" is requested by the client via a SYNC or PING 
request. Nothing we can do to force it.

Also, please use the appropriate mailing list to ask for further 
support since this is not a bug or feature request.

http://www.horde.org/mail/ contains a list of all available mailing lists.

After upgrading my Nexus phone to Android 5, I recognized that - due 
to the usage of self-signed certificates - my user account was deleted 
after the update process.
So after re-adding my user account, I noticed, that Android started 
syncing mails, contacts and calendar items as intended.
However, just after finishing the last entry, Android decides to 
delete everything due to the following error:

I/Exchange( 3186): PSS syncEnd for account acct:2
I/Exchange( 3186): PSS push enabled is unknown
I/Exchange( 3186): PSS no push enabled acct:2.
I/Exchange( 3186): PSS removing account state for acct:2
I/Exchange( 3186): PSS removed last account; stopping service.

The syncing process starts again and again (...) without ever finally 
succeeding.
I know that this issue is not with Horde, since there is no error in 
the Horde log and it worked before just fine. However, I was asking 
myself, whether it is possible to "enable push services". I was always 
under the impression that push was already working with Horde.
Does anyone have the same issue or a hint how to proceed?

I am using Horde Groupware 5.2.3 with Horde_ActiveSync 2.20.0