Jordan curves admitting only acyclic inscriptions of squares

The (recently solved) inscribed square problem or Toeplitz conjecture posits that every closed, plane continuous (Jordan) curve $${\it \Gamma}$$ in $$\mathbb{R}^2$$ contains all vertices of some square. It appears this theorem was just proven!

Most examples resemble the one on the left, in which the natural continuous parameterization along $${\it \Gamma}$$ intersects the square's points in sequence. Let's call such an inscription cyclic. However, some Jordan curves contain the points of a square in non-sequential order, such as shown at the right. Let's call such an inscription acyclic.

Questions

• Are there Jordan curves that admit only acyclic inscriptions (and not also cyclic inscriptions)?
• Given the inscribed square problem was just answered in the affirmative, can one prove whether acyclic-only curves exist?
• Alternatively, or additionally: can one provide an example of such an acyclic-only curve?

My conjecture is that there are no such acyclic-only Jordan curves. My first approach has been to assume that there is a given acyclic square inscription for a Jordan curve and then prove--invoking continuity assumptions and topological methods--that there must also be a cyclic inscription. Alas, such a proof for even this partial case has been elusive.

• Each edit bumps this post to the top of the active list and another question off. Could we perhaps wrap up the editing at least for a while, so that people have a stable question to look at? Jul 15 '17 at 3:57

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.