Do suction cup brush caddies adhere to lacquered cabinet doors?

[Caslon]

Do lightweight plastic brush and sponge caddies with suction cups stick to the inside of the under sink cabinet door? The door surface is natural wood with lacquer finish. It's a leased condo and I would be charged for damage if I screwed it on. Most reviewers of the product have them using it for the sink, not a door surface.

The only other alternative I can think of would be to use those 3M adhesive pads that remove cleanly (the ones for hanging light weight pictures). I would somehow fashion those on to the cup holder, then attach the caddy to the door that way.

 

[Caslon]

A better question to pose might have been, where do you keep your kitchen sponges and wash brush? I'm not going to buy those holders that attach to the faucet or sink!
I found an inexpensive brush and sponge caddy, but the suction cups do NOT stick to lacquered finishes. I will try the 3M removable adhesive strips, but I need to cut off some
plastic on the holder for it to fit flush against the cabinet door. Stay tuned for this exciting development, ha.