Full pattern support conjecture for skew Brownian permutons

Let $\bm \mu_{\rho,q}$ be a skew Brownian permuton with parameters $(\rho,q)\in(-1,1)\times(0,1)$. For $k\in\mathbb{Z}_{>0}$, let $\pi$ be a permutation pattern of size $k$, and let $\operatorname{Perm}_k(\bm \mu_{\rho,q})$ be the permutation induced by $k$ independent points sampled from $\bm \mu_{\rho,q}$. Full pattern support conjecture. For all $(\rho,q)\in(-1,1)\times(0,1)$, all $k\in\mathbb{Z}_{>0}$, and every pattern $\pi$ of size $k$,

$$\mathbb{P}(\operatorname{Perm}_k(\bm \mu_{\rho,q})=\pi)>0.$$

This asserts that every finite permutation pattern has positive probability for interior parameter values. The source presents it as an intuition-based statement; its resolution is not specified in the supplied text.