Hello,
I'm 100% sure it uses the same mechanism as if you use the interior button. If it starts to close the passenger window, the electric module should also be the same.
What happens when you close the driver side window before you leave the car (with the button) and then use the module to close only the passenger window?
Might be that the battery is too weak. Can you connect a multimeter to the battery terminals and observe the voltage when the windows are closing? What are the readings?