I am not sure whether my comment would be seen on such an old thread, but recently, I experienced similar issue. There's end-to-end ssl connectivity from user to SAP servers through F5. It's configured with source IP address as affinity method. The sessions were getting timed out prematurely when left idle. I increased the persistence timeout and it resolved the issue.
What I do not understand is are as following :-
1. Why increasing the time out settings for idle time out and configuring keep-alive did not maintain the session ?
2. Now when the user session is maintained for over an half-an-hour, I do not see it's connection logs on F5 tmsh using show sys connection command.