Asymptotic enumeration conjecture for deque-sortable and parallel two-stack-sortable permutations

For $n\in\mathbb{Z}_{\geq0}$, let $p_n$ be the number of permutations of size $n$ sortable by two stacks in parallel, and let $d_n$ be the number sortable by a deque. The generating functions $D$ and $P$ satisfy the equivalent relations stated in the preceding theorem.

Asymptotic enumeration conjecture. There exist constants $\mu$ and $\gamma$ such that

$$p_n\sim \operatorname{const}\cdot\mu^n\cdot n^\gamma$$

and

$$d_n\sim \operatorname{const}\cdot\mu^n\cdot n^{-3/2}.$$

This conjecture is motivated by the relation between the generating functions for the two permutation classes. The paper presents it as a stronger conjecture whose proof is reduced to conjectures about the quarter-plane-loop generating function $Q(a,u)$; its resolution is not established here.