CARP HASH work for HTTPS traffic

what information do you want to persist on?

if layer 7 information such as uri, ssl offloading is also required.

Not Layer 7, Layer 4 like Source_ip, And when persisted without timeout, is it going to be persisted throughout a session to the same server?

when persisted without timeout, is it going to be persisted throughout a session to the same server?

yes as long as pool member is up.