Longest increasing subsequence exponent conjecture for skew Brownian permutons

For $(\rho,q)\in[-1,1]\times(0,1)$, let $\bm{\mu}_{\rho,q}$ be the skew Brownian permuton. Let $\operatorname{Perm}(\bm{\mu}_{\rho,q},n)$ be a sample of size $n$, and let $\operatorname{LIS}$ denote its longest increasing subsequence length.

Skew Brownian LIS exponent conjecture. There exists a function $\ell:[-1,1]\times(0,1)\to[1/2,1)$ such that, with probability tending to $1$ as $n\to\infty$,

$$\operatorname{LIS}(\operatorname{Perm}(\bm{\mu}_{\rho,q},n))=n^{\ell(\rho,q)+o(1)}.$$

Moreover, $\ell(-1,q)=1/2$, $\ell(1,q)=d(1-q)\in[\alpha_*(1-q),\beta^*(1-q)]$, and $\ell(\rho,q)$ is continuous, non-increasing in $q$, and non-decreasing in $\rho$. The conjecture extends the Brownian separable-permuton exponent problem to the full skew Brownian family; only bounds and special cases are currently available.