I would assume the answer to that question is "no", because for an IMAP client there is no reason or need to send a copy of your emails or even the metadata to Google, and the shame and suspicion such a move would cause if they were caught doing it would be nontrivial. Not to mention, checking for it is not that hard.

One way to test this though would be to setup outbound filtering (see my previous commments), and to use the always-on VPN functionality on your phone to shunt your traffic through said filter. This would allow you to see if any of your emails are being exfiltrated without your knowledge.

I haven't tested this yet myself.