Why create an issue when none exists? If one is to do it and there is no talking them out of it, the insulation and vapor barrier should be behind it, preference goes to sheet foam or spray foam. One could also sacrifice a couple of inches and build a secondary wall on the warm side of the outside wall for the mechanical, preference would be to allow for some air movement in this wall.
At my first house I had a laundry sink installed on an outside wall, I sheet foam insulated the stud cavity behind it as well as either stud cavity adjacent. I then installed 3"x10" return air grills in the drywall at the bottom and top of the chase, which allowed for warm air movement through the space.
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"The trouble with people idiot-proofing things, is the resulting evolution of the idiot." Me
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