Humidity control is good, but it won't do much about the heat factor.

I'd find out what the actual ambient heat buildup is inside that closet before commiting to anything.

Fast way, put a thermometer in there in the morning and then come by and check it after the heat of the noonday, around the 1 to 2 PM mark. If it is much above the 77 degree F mark, I'd rethink.
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