Here are suggestion:
- Running HCW to create connector to make sure the mail flow between Exchange online and Exchange on-premises.
- Don't enable centralized mail transport, make sure Exchange online send email out from Exchange online directly.
- Remove send connector which used to send emails to the Internet from Exchange on-premises.
Here is an article about mail flow in hybrid, if you don't enable centralized mail transport, Exchange online will send email to the Internet directly, Exchange on-premise will send email to Internet from the local environment(if you don't create related send connector on the Exchange on-premises, Exchange on-premises user will cannot send email to the Internet), it doesn't effect the mail flow between Exchange online and Exchange on-premises.
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