Grothendieck–Plancherel conjecture for the homogeneous limit shape

Let $x,y\searrow 0$ and let $\beta=\beta(x)\nearrow 0$ satisfy $-\beta(x)\ll x$. Let $\mathfrak{W}(u\mid x,y,\beta(x))$ denote the Grothendieck limit shape, and let $\Omega(u)$ be the Vershik–Kerov–Logan–Shepp shape. Grothendieck–Plancherel conjecture. The rescaled limit shape

$$\frac{1}{\sqrt{xy}}\mathfrak{W}(u\sqrt{xy}\mid x,y,\beta(x))$$

converges to $\Omega(u)$. This is motivated by the fact that the bounding Schur-process limit shapes become close to the Plancherel limit shape in this regime; the conjecture is supported by numerical experiments, but no resolution is given here.