Problem: for a fixed integer $m\geqslant 3$ find all $n$ such that no $n$-gon can be dissected into convex $m$-gons.

I would be very grateful for any information on this problem.

Remark 1. There some results on this problem known from mathematical competitions. For example, a square cannot be dissected into convex hexagons, and 7-gon cannot be dissected into convex hexagons.

Remark 2. I with my student Bohdan Kivva have solved the problem. However, the problem seems to be classical, so I have great doubts about originality of this result. Searching the Google Scholar didn't give any results.