Redundant LDAP set-up

I am trying to establish LDAP as authentication source for Gitea.
It works mostly as expected using the good guide found here:

I have a pair of FreeIPA servers operating in a replicated and redundant pair. I have configured Gitea to use each FreeIPA server as an authentication source, using the Admin GUI.
Both FreeIPA servers are set up and confirmed to be working individually (one at a time).
However, the goal is to have a High availability set-up.
I have tested taking one of them out of service, trying to get users automatically authenticated via the other , that holds the same replicated data.
I cannot get this to work. It seems that each user can only be authenticated via the LDAP source where it was logged in the first time.
I was expecting the client (Gitea) to try the next available auth source, if the first choice does not answer.
Any idea what I am doing wrong?