Jordan curves admitting only acyclic inscriptions of squares

The recent work by Jason Cantarella, Elizabeth Denne and John McCleary implies (but does not prove) that the answer to all questions is negative, as I conjectured. Specifically, under certain general conditions one can find a cyclic inscription, and thus precluding the existence of an acyclic-only Jordan curve.

I am not a member here and so could not provide a comment. The claim that it has recently been solved is inaccurate. Green and Lobb solve the $\textbf{smooth}$ version of the Rectangle Peg Problem. For the square case, that’s been known since about a 100 years now (I suppose Schnirelman was the first) or at least some decades now. Indeed, it is true for all continuously differentiable curves.

I suppose you can check a few results of Richard Schwarz, in the last couple of years, which may answer your questions regarding the (a)cyclicality.