Login session on AWS ECS

We are trying to set up WP on ECS with two services and a Load Balancer. So far, everything works great, except logging in is impossible. Authentication cookies disappear as soon as they are created.

We cannot enable sticky sessions because doing so would defeat the purpose of having a load balancer. If you are wondering why, users will be bound to their initial service, which can overload the service.

Do you have any recommendations or ideas about which direction we could take for our next steps? Thanks.