The biased-separable representation conjecture for skew Brownian permutons

For $\rho\in[-1,1]$ and $q\in[0,1]$, the skew Brownian permuton $\bm\mu_{\rho,q}$ is the push-forward of Lebesgue measure on $[0,1]$ by the map $(\operatorname{Id},\varphi_{\bm{\mathscr Z}_{\rho,q}})$. For $p\in[0,1]$, let $\bm\mu^{(p)}$ be the biased Brownian separable permuton.

Biased-separable representation conjecture. For every $p\in[0,1]$,

$$\bm\mu^{(p)}\stackrel{d}{=}\bm\mu_{1,1-p}.$$

The Baxter permuton is already identified with the case $(\rho,q)=(-1/2,1/2)$. This conjecture identifies the opposite extremal family with the biased Brownian separable permutons and provides evidence for the broader skew Brownian universality picture.