Let $Fr(X)$ be the frontiere of any $X$ subset of the plane.
Does there exists $A,B$ convex compact sets of the plane, such that $C:=Asetminus B$ is connex and not empty, and such that there is no square inscribed $Fr(C)$ that has a vertex in $Acap Fr(C)$ ?
If the answer is no, this would kind of mean that the Toeplitz conjecture is kind of linked to convexity (one of the firts intersting resolved case was the case of the frontiere of a convex)
If the answer is yes… then it suggests a “fractal” way to built some counterexample.
Do not hesitate to say your “pronostic” in the comments, even if it is not argued so far, I have absolutally no intuition myself about the pronostic (I would say it must be “YES” but I would be influenced by the fact that according to me, if it is “YES”, then there is a counterexample not far, and this would have been found – but I might as well underevaluate the difficulty of “YES=> counterexample” …)