Grothendieck limit-shape conjecture in the large negative-beta regime

Let $x,y\searrow 0$ and let $\beta=\beta(y)=-K/y$, with fixed $K>0$. Let $\mathfrak{W}(u\mid x,y,\beta(y))$ be the Grothendieck limit shape, and let $\Omega$ be the Vershik–Kerov–Logan–Shepp shape. Large negative-beta Grothendieck conjecture. There exists $K_0>0$ such that, for all $K>K_0$, in the $O(\!\sqrt{xy}\!)$-neighborhood of $u=1$, the curve $\mathfrak{W}(u\mid x,y,\beta(y))$ is close to

$$\frac{u+1}{2}+\frac{\sqrt{xy}}{2}\Omega_{(K)}\left(\frac{u-1}{\sqrt{xy}}\right),$$

where $\Omega_{(K)}$ is a suitable $K$-dependent deformation of $\Omega$; the source further expects that as $K\to-\infty$, the shapes $\Omega_{(K)}$ approach $\Omega$. This conjecture concerns the Plancherel-like curved part observed near the frozen staircase facet, and is supported by numerical experimentation.