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One of the support boards did become unglued when I drilled, as I expected. But the other one seems to be enough to keep the drywall in place.
I am not going to win any awards, but at least you can’t casually shove this part of the wall in. What more can you expect?
(While I was pushing, I was afraid. Despite me telling myself that this hole doesn’t matter in the greater scheme of things, I would have been crushed if the drywall collapsed in.)