The Cayley-equivalence maximum conjecture

Let $p$ and $q$ be Cayley permutations of the same length, and let $\max(p)$ denote the largest letter occurring in $p$. Write $p\sim_c q$ for Cayley-equivalence. The Cayley-equivalence maximum conjecture. If

$$p\sim_c q,$$

then

$$\max(p)=\max(q).$$

The conjecture asks whether Cayley-equivalence forces the two patterns to have the same maximum letter. The source observes that the analogous conclusion is immediate for content-multiplicity equivalence, but leaves the Cayley-equivalence case as an open problem.