The Cayley-equivalence content conjecture

Let $p$ and $q$ be Cayley permutations of the same length. Write $p\sim_c q$ for Cayley-equivalence and $p\sim_{cm}q$ for Cayley-equivalence with respect to content multiplicities. The Cayley-equivalence content conjecture. If

$$p\sim_c q,$$

then

$$p\sim_{cm}q.$$

The conjecture is posed as an open problem. The paper identifies $p=13442$ and $q=14233$ as the smallest candidate for a counterexample and notes that their relevant Cayley-avoidance counts agree through size $9$, while a content-multiplicity count already differs at the indicated parameters. No proof or counterexample is supplied.