Finite-size correction conjecture for longest increasing subsequences

Let $l_N^\square$ be the length of the longest increasing subsequence in the finite $N$ model, and let $E_2^{\rm soft}(0;(t,\infty))$ denote the soft-edge limiting distribution. Set

$$F_{2,0}(t)=E_2^{\rm soft}(0;(t,\infty)).$$

Here $t^*$ is the rescaled variable defined in the source's equation (1.1i). Finite-size correction conjecture. For some function $F_{2,1}(t)$,

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

This conjectures the first finite-$N$ correction to the limiting soft-edge law. The preceding discussion states that the available Poissonized asymptotics do not determine this correction and that the formula is supported by numerical methods.