Tracy–Widom crossover conjecture for rectangular Young tableaux

Let $(a_n)$ be a nondecreasing sequence with $a_n\to\infty$ as $n\to\infty$, and let $X^{(n)}_{i,j}$ denote the relevant entry of the random rectangular Young tableau. Write $TW$ for the Tracy–Widom distribution. Tracy–Widom crossover conjecture. Up to multiplicative constants, both limits hold:

$$\frac{a_n^{1/6}\bigl(X^{(n)}_{\floor{a_n},n}-{\mathbb E}X^{(n)}_{\floor{a_n},n}\bigr)}{n^{3/2}}\stackrel{\mathrm{law}}{\longrightarrow}TW$$ $$\frac{X^{(n)}_{\floor{a_n},1}-{\mathbb E}X^{(n)}_{\floor{a_n},1}}{a_n^{4/3}}\stackrel{\mathrm{law}}{\longrightarrow}TW.$$

The conjecture is intended to describe the transition between the deterministic regime at the corner and the fluctuation regimes along the boundary and at the opposite edge. The source gives no resolution, so its status remains open.