Convergence to the Schubert permuton

For each $n$, let $\Upsilon_w=\mathfrak{S}_w(1^n)$ for $w\in S_n$, and define the Schubert measure on $S_n$ by

$$\operatorname{\mathbb{P}}(w)=\frac{\Upsilon_w}{\sum_{v\in S_n}\Upsilon_v}.$$

A permuton is a probability measure on $[0,1]^2$ with uniform marginals.

Convergence-to-the-Schubert-permuton conjecture. The random permutations distributed according to this measure converge, as $n\to\infty$, to a deterministic permuton $\mu$ on $[0,1]^2$, called the Schubert permuton, supported inside a cone and having a singular component along the southeast boundary curve.

The conjecture is motivated by numerical observations and by analogous results for Grothendieck random permutations. The source does not identify the limiting measure or prove convergence, so the claim remains open.