You are right about Connectors support inside projects. Just my assumption was wrong.
I have used Connectors in Projects for about 2 months and it just stopped working about week ago - at the time when Connectors stopped working in Projects, some members posted problems with their Connectors; I thought it may be corelated to my observations, here is the link to original post: https://community.openai.com/t/chatgpt-not-using-connected-local-app-sdk/1369390