Increasing-pattern extremal conjecture for distant-pattern avoidance

Let $m\geq3$ and $r\geq1$, and let $q$ be any classical pattern of size $m$. Increasing-pattern extremal conjecture. There exists $n_0\in\mathbb{N}$ such that for every natural number $n>n_0$,

$$|\operatorname{Av}_{n}(\operatorname{dist}_{r}(12\cdots m))|\geq|\operatorname{Av}_{n}(\operatorname{dist}_{r}(q))|.$$

Thus, among classical patterns of size $m$, the increasing pattern is conjectured to have asymptotically largest distant-pattern avoidance classes for each fixed gap size $r$. The claim is listed among the paper's open problems and is unresolved in the supplied source.