The RBPD limit-shape and arctic-curve conjecture

Let $h$ be the height function of a uniformly random reduced bumpless pipe dream (RBPD) of size $n$. A limit shape is a deterministic limiting height function; an arctic curve is the deterministic boundary separating frozen and liquid regions.

RBPD limit-shape and arctic-curve conjecture. As $n\to\infty$, the height function converges to a deterministic limit shape consisting of four frozen regions adjacent to the domain's corners, where the height function is linear, and a liquid region, where it is curved, separated by a deterministic arctic curve. The southeast arc of this arctic curve coincides with the support of the singular component of the Schubert permuton.

The conjecture is based on simulations of the uniformly random RBPD. The paper observes symmetry about the main diagonal but no apparent quarter-turn symmetry, and does not establish either the limit shape or the asserted identification of the southeast arc.