Conjectured asymptotic expansion for variances of longest monotone subsequence lengths

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

$$\operatorname{Var}(L_n^{\varoast})=\sum_{j=0}^{m}\nu_{\varoast,j}(\gamma n)^{(2-\gamma j)/6}+O\bigl(n^{(2-\gamma(m+1))/6}\bigr)\bigg|_{\gamma=\gamma(\varoast)},$$

where the coefficients $\nu_{\varoast,j}$ can be expressed in terms of the $\mu_{\varoast,k}$ and the second moments of $F_{\beta(\varoast)}'$ and $F_{\varoast,j}^*$. This conjecture extends the proposed expectation expansion to the fluctuations of the longest monotone subsequence lengths; the source does not provide explicit formulas for all $\nu_{\varoast,j}$.