This is talking about the playground then. I also noticed that the selection is not remembered as a playground setting just switching between assistants. It is a header you send in the API request, not an assistant setting with any API “remember” method.
You leave “chat”, it doesn’t remember your settings or prompt either, so this is somewhat expected. The Playground is more a demonstrator, and promoting the latest method is understood. Maybe a “super-switch” such as a drop-down v1 or v2 for two different playground views for any assistants then selected would have been a better method than a little callout within.