The Cayley-avoidance maximum inequality conjecture

Let $p$ and $q$ be Cayley permutations of the same length, and let $\max(p)$ denote the largest letter occurring in $p$. Let $\operatorname{Cay}_n(p)$ be the set of Cayley permutations of size $n$ avoiding $p$. The Cayley-avoidance maximum inequality conjecture. If

$$\max(p)\leq\max(q),$$

then, for every relevant size $n$,

$$|\operatorname{Cay}_n(p)|\geq|\operatorname{Cay}_n(q)|.$$

The conjecture predicts that a pattern with no larger maximum has at least as many Cayley-permutations avoiders as a pattern with a larger maximum. The paper states this as an open problem; no proof or counterexample is reported.