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.