| Unless you replace the door with a solid core, and add at least 1 layer of gypsum to the entire room, including the ceiling, you really aren't going to stop one bit of sound from exiting the room.
Your plan for a window plug will knock a bit of sound down, but very, very little. It should at least contain some sort of frame that you can attach a bit of gasketing to it.
Remember, where air goes, so goes sound.
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