Unfortunately it did not. I've tried for many times. It never asked me close any of the doors. It locks the gear, tells me to release parking brake and starts pushback with the doors open.
I can only repeat and confirm that GSX will ask your to close cargo doors during Pushback, before you have been asked to release the Parking Brakes.
But of course, you are using an airplane that you configured to IGNORE a door status, either because you made it like that, or you downloaded a file made by someone that configured it that way, GSX is doing exactly what you told it to do: ignoring the door status so, it will be then entirely up to you to close it, since you asked GSX to ignore the door.