There could be different opinions whether Push should be posting notifications even when the app is open/active.
But in either case, I agree with Paris Geek, the expectation should be clear for the user.
At present if I were to disable "post notifications even when the app is open", I'd expect that to be for ALL notifications, push or not.
Actually, I think that if there is an option to disable new message notifications (while the app is active), that should be applicable to all of them, - scheduled, on event, push. For a user, - it is a new message notification, regardless what technology produced it. (And for many (most?) users, Push = scheduled with extremely small interval) Why should it be different for scheduled sync and push? The user either wants it when looking for something else in the app, or doesn't want.
Personally, I want those notifications at all times, so, I've always had them enabled.