The non-subtransitive cyclic-quadrilateral conjecture

A cyclic polygon is a polygon whose vertices lie on a common circle. A finite set is subtransitive if it is congruent to a subset of a finite transitive set.

Cyclic-quadrilateral conjecture. There exists a cyclic quadrilateral that is not subtransitive.

The paper proves that almost every cyclic $16$-gon is not subtransitive, but does not give an explicit example. It therefore asks for an explicit non-subtransitive cyclic polygon and conjectures that four vertices already suffice.