GCP - Identify which IAM user executes a Cloud SQL connection in the logs

Question

We would like to identify which IAM user executes a Cloud SQL Proxy connection in the logs. However, we can see this message:

  "textPayload": "2022-11-14 10:41:29.849 UTC [358324]: [1-1] db=[unknown],user=[unknown] LOG:  connection received: host=[local]",

We have the following flags enabled in the Postgres database:

• cloudsql.enable_pgaudit = ON
• pgaudit.log = all
• log_connections = On
• log_disconnections = On

We would like to know what other missing properties we need to set to have that "db" and "user" fields populated properly.

Extra info: https://medium.com/google-cloud/correlate-statement-logs-in-cloudsql-for-postgres-with-connection-sessions-5bae4ade38f5

Thanks.