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There could be several reasons why Flask sessions are not being maintained on Google App Engine in a basic chess game:

1. The session configuration is not set up correctly. Make sure that the SECRETKEY is set to a long random string and SESSIONCOOKIE_SECURE is set to True to prevent session hijacking.

2. The session cookies are being blocked by the browser's third-party cookie policy. To ensure that the session cookies are not blocked, enable SameSite=None attribute and include secure flag in session cookies.

3. The server may be running on multiple instances, and each instance is not sharing the same session data. In this case, we need to use a central datastore or a shared cache system like Memcached to store the session data.

4. The session data may be expiring too quickly. Make sure that the SESSION_TIMEOUT configuration is set to a reasonable value.

5. There may be an issue with the implementation of the session storage system. Try using a different storage system, like Redis, to see if the issue persists.