That closet design is too small for a full-size washer and dryer (typical width 2 x 27” = 54”). You’d have to build the clothes racks and pocket door after the units are installed, and dismantling the same would be required for future replacement. You’ll lose more than half of the closet space in the room.
The sketch shows a front door on the washer, unlike the photo, so maybe you’re planning a change. Our current front loader is our first experience with that style. It spins at very high speed to dry the clothes as much as possible and the vibration is noticeable in the floor. Ours is an unusually thick floor on closely spaced joists so the vibration isn’t a problem, but that wouldn’t be the case for many installations. The manufacturer recommends installing on concrete.
Have you considered the difficulty of running new wiring/gas/water to the third floor location and how to duct the dryer exhaust to outside?
Since you asked our opinions, I’d say that if you’re younger stick with the machines downstairs where they are less trouble and you get more closet space. If you’re old enough to be retired and don’t need that much closet space anyway, lose the difficulty of moving laundry and downstairs by installing in the closet.