The corner permuton limit conjecture for almost square permutations

Let $m=n+k$, and let $\bm\sigma_n^k$ be uniform among permutations of size $m$ with exactly $n$ left-to-right minima and $k$ internal points. For $\alpha>0$, let $\nu_*^\alpha$ be the permuton supported on the bottom-left corner shape defined by the curve and measure construction preceding the conjecture.

Corner permuton limit conjecture. If $k,n\to\infty$ with $k/n\to\alpha\in\mathbb{R}_{>0}$, then

$$\mu_{\bm\sigma_n^k}\stackrel{d}{\longrightarrow}\nu_*^\alpha.$$

This is a simplified model motivating the phase-transition conjecture for almost square permutations, isolating the contribution of left-to-right minima and internal points.