Conjectured asymptotic expansion for expected longest monotone subsequence lengths

Let $L_n^{\varoast}$ denote the relevant longest monotone subsequence length for $\varoast\in\{\boxslash,\boxbslash,\boxdot\}$, let $\gamma(\varoast)$ and $\delta(\varoast)$ be the scaling and centering constants, and let $F_{\beta(\varoast)}'$ and $F_{\varoast,j}^*$ be the coefficient functions defined in the source. Expectation expansion conjecture. For every fixed non-negative integer $m$, as $n\to\infty$,

$$\mathbb E(L_n^{\varoast})=2\sqrt{\gamma n}+\delta(\varoast)+\sum_{j=0}^{m}\mu_{\varoast,j}(\gamma n)^{(1-\gamma j)/6}+O\bigl(n^{(1-\gamma(m+1))/6}\bigr)\bigg|_{\gamma=\gamma(\varoast)},$$

where

$$\mu_{\varoast,0}=\int_{-\infty}^{\infty}tF_{\beta(\varoast)}'(t)\,dt,\qquad \mu_{\varoast,j}=\int_{-\infty}^{\infty}tF_{\varoast,j}^*(t)\,dt\quad(j=1,2,\ldots).$$

This conjecture would give asymptotic expansions for the expectations in all three involution symmetry classes. It is motivated by inserting the probability-density expansions and replacing the resulting trapezoidal sums by integrals under exponential decay and analyticity assumptions.