3/4 in high quality ply. Enough for the two end pieces and I used two about 2" x about 38" pieces as stretchers.
Shaper Origin
1/4" upturn spiral bits.
1" dowel 48" long.
Safety pins to hold dowel, see photo.
4 screws about 1 1/2 long for 3/4 in stretcher attachment.
Starting with the end pieces, cut the rabbet out for the stretcher along the back under the wall cleat to a depth of ~1/4. Cut the holes to the correct diameter to allow the dowel to set snuggly but not real tight. Then cut out the cleats, recommend going down with the finish cut leaving a few 1,000ths then a final pass leaving tabs, break out and sand.
Set length of the stretchers and dowel based on width of paper roll. Mine was 36" so the distance between the two ends was cut at about an inch longer to allow for some space so the paper roll.
Did a butt joint for the back stretcher, so it will be longer longer than a stretcher along the bottom as well. Both seemed necessary for stability. Glued and screwed the stretchers.
Used hardware store "safety pins" to keep the down from sliding out from either side, but allow for paper changes. Drill hole in dowel for pins.