Two doors open toward each other, and one will rub along the other. What can I put on one (or both) to protect them over time?

Question

I’m installing an Ikea Pax cabinet next to an existing bathroom door. The doors open toward each other -- not ideal, I know, but necessary to maximize the space in this old, small apartment -- and a corner of the Pax door will rub along the top rail of the older door, as shown here. (The rest of the older door is slightly off-vertical so it doesn't make contact with the Pax.) What can I put on the Pax’s corner (after I paint it) that’s not too visible but will protect the doors over time? UHMW tape like this seems a good option, but I’ve never used it and I’m wondering if that or something else is the best solution.

Ideally, whatever I use won’t slowly buff the older door’s built-up paint to a nice shine over time (they won’t be constantly rubbing but we also won’t be constantly paying attention). Or should I pre-emptively put something on the older door as well? Again, a strip or 2 of low-friction, translucent tape seems to make sense, but I’d welcome all suggestions.

Thanks in advance!

Edit: Eventually I'll learn that what's clear to me is invisible to the rest of the world. 🫤 Only that one ~4" section of vertical corner on the (taller) Pax door makes contact, which can be anywhere along the very top rail of the bathroom door. This is outside the bathroom, which is why I'd like a less-visible approach. The cabinet will be for tools and such so it'll stay open when I'm working. The bathroom door also has to stay at least partly open since the cat's litter box is in there (another joy of apartment living).

Answer 1

Low tech solutions:

Put a towel rail, the widest one that will fit, on the back of the entry door as high as possible. Always leave a towel there. In a small bathroom I'm surprised that's not already crossed of the list. Anway, the cupboard door will rub on the towel, and if you get it right there will be no buffing.

If you don't want a towel there, get a strip of felt with a peel-off sticky back and stick it along the entire top of the door. Get a rectangle of hard teflon or nylong, about half an inch wide and 4 inches high, also stick-on, and stick it on to the cupboard door where it will make contact with the felt. You can reverse these if you want.