The circular limit-shape conjecture for totally symmetric self-complementary plane partitions

A totally symmetric self-complementary plane partition (TSSCPP) of size $n$ has vertices $(x_1,x_2)\in\mathtt{V}_n$. Rescale it so that its three corners converge to $(-2,0)$, $(0,0)$, and $(-2,-2/\sqrt{3})$ as $n\to\infty$, with

$$x_1=[(X+2)n],\qquad x_2=[(\sqrt{3}Y+2)n].$$

Circular limit-shape conjecture. As $n\to\infty$, the limit-shape curve is

$$X^2+Y^2=4,$$

and the region $X^2+Y^2\geq 4$ in the rescaled TSSCPP is frozen.

This conjecture describes the macroscopic frozen boundary of a uniformly random TSSCPP. The surrounding discussion reports nonrigorous computations suggesting this limit shape, together with Airy-kernel statistics at the edge; no proof or resolution is supplied here.