Exchange 2013 session affinity. Once again.

I'm confused as to why you say that you don't need a persistence profile. My understanding is that for OWA the user authenticates to the server via either forms or NTLM and accesses their mailbox from that server - login details are not shared between CAS servers as far as I understand. Have you tried doing a tcpdump to see what is actually happening? If not then it might be worthwhile. I should imagine that what is happening is that, with no persistence, the user is starting a session and authenticating to a server, then being loadbalanced to a different server on the next session and is asked to sign in again.