Variation of the inscribed square problem

Question

The inscribed square problem (summary here) is currently open:

Does every Jordan curve admit an inscribed square?

(It is not required that the vertices of the square appear along the curve in any particular order)

I failed to come up with a curve that needs the precision about the order of the vertices. Is there a trivial example? Or is it that we simply do not care?

Answer 1

Let me preface this with the fact that I am unfamiliar with this problem or attempts at proving/disproving the conjecture.

I can't include a picture in a comment so I post this here...

Here's an attempt at drawing a curve such that you can't have a square appear along the perimeter with its vertices "in order". I can't see any other squares that match up with my curve, but then again maybe I missed something.

Thanks for pointing out this interesting conjecture. :)

Answer 2

one more

Oops! Your answer couldn't be submitted because: Body must be at least 30 characters; you entered 9.