Connecting to Session Pooler in self-hosted instance results in FATAL: Tenant or user not found.

I have the questions described here: https://github.com/orgs/supabase/discussions/40297

I configured everything via the .env files, but when I try to connect via the session pooler, I get the error:

[XX000] FATAL: Tenant or user not found.

I tried to add the tenant to the username like this:
<user>.<POOLER_TENANT_ID>

still the same problem...

is it possible to use the session pooler in the self-hosted instance? by configuring it via the .env vars?

what would be the connection string if I use the plain .env variables from the documentation: https://github.com/supabase/supabase/blob/master/docker/.env.example
and the stock docker compose file https://github.com/supabase/supabase/blob/master/docker/docker-compose.yml ?
GitHub
I’m running a local Supabase instance using the official Docker Compose setup Everything starts fine, but when I open the project settings page for the Session Pooler connection string at https://[...
GitHub
The Postgres development platform. Supabase gives you a dedicated Postgres database to build your web, mobile, and AI applications. - supabase/supabase
GitHub
The Postgres development platform. Supabase gives you a dedicated Postgres database to build your web, mobile, and AI applications. - supabase/supabase
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