Keeping a session "sticky" once it is assigned to a server is a somewhat different issue than controlling the assignment.
"Persistence" is more what I want but digging this out of HAProxy configuration and making changes on the fly looks troublesome.
https://www.haproxy.com/blog/enable-sticky-sessions-in-hapro... https://www.haproxy.com/blog/load-balancing-affinity-persist...