The limit-shape conjecture for the MERW pyramidal process

Let $(P_n)_{n\geq 0}$ be the suitably scaled maximal entropy random walk pyramidal process. Define

$$\Omega(u)=\frac{2}{\pi}\left[u\arcsin(u)+\sqrt{1-u^2}\right],\qquad -1\leq u\leq 1,$$

and let $\mathcal{H}$ be the graph of $\Omega$. Set

$$T(u,v)=\left(\frac{u+v}{2},\frac{v-u}{2}\right),\qquad S(x,y)=(-x,y),$$

and $\mathcal{G}=T(\mathcal{H})\cup ST(\mathcal{H})$. Limit-shape conjecture. The distance between the boundary of $P_n$ and $\mathcal{G}$ tends to zero with probability one. This proposes that the scaled MERW pyramidal process has the same symmetric limit shape as the classical Plancherel growth process. The statement is supported by numerical simulations in the paper, but no proof or resolution is supplied.