Finite-size correction conjecture for symmetric longest increasing subsequences

Let $l_N^{\boxbslash}$ and $l_N^{\boxslash}$ denote the lengths of the two symmetric longest increasing subsequences considered in the source, and let $E_1^{\rm soft}(0;(t,\infty))$ and $\tilde E_4^{\rm soft}(0;(t,\infty))$ be their respective soft-edge limiting distributions. Let $t^*$ be the rescaled variable defined in the source's equation (1.1i). Symmetric longest-increasing-subsequence correction conjecture. Set

$$F_{1,0}(t)=E_1^{\rm soft}(0;(t,\infty)).$$

For some $F_{1,1}(t)$,

$$\Pr\left(\frac{l_N^{\boxbslash}+1-2\sqrt{N}}{N^{1/6}}\leq t\right)=F_{1,0}(t^*)+\frac{1}{N^{1/3}}F_{1,1}(t)+\cdots.$$

Similarly, set $F_{4,0}(t)=\tilde E_4^{\rm soft}(0;(t,\infty))$. For some $F_{4,1}(t)$,

$$\Pr\left(\frac{l_N^{\boxslash}-1-2\sqrt{N}}{N^{1/6}}\leq t\right)=F_{4,0}(t^*)+\frac{1}{N^{1/3}}F_{4,1}(t)+\cdots.$$

These formulas conjecture the first finite-size corrections for the two symmetric longest-increasing-subsequence distributions. The source says that simulations and numerical plots provide evidence for this analogue of an earlier conjecture, while no analytic derivation is given.