You are trying to go at least 5 high? but how many wide? Let's say 10 wide. Set out your first set 10 wide. Now lay a piece of thin cardboard on top of that row full 10 width. Keep building. The cardboard will spread the load, and also lock each piece together.

Basically, you are no longer stacking 5 pops directly on top of each other. The cardboard layers kind of act like concrete between layers of bricks. (not that Pops are bricks! lol)

Wanna take it further? Allow a 1/2" flap for each layer of cardboard. Thumbtack that flap to the wall behind each layer. NOW, that piece of cardboard can't pull away from the wall.

This may sound like a lot; but you could probably pull it off doing every other layer of Pops.