The semi-Baxter and strong-Baxter universality conjecture

A semi-Baxter permutation and a strong-Baxter permutation are permutation classes encoded by two-dimensional labeled generating trees. Let $\mu_\sigma$ be the permuton associated with a permutation $\sigma$, and let $\bm\mu_{\rho,q}$ be the skew Brownian permuton.

Semi-Baxter and strong-Baxter universality conjecture. The permuton limit of uniform semi-Baxter or strong-Baxter permutations is the skew Brownian permuton. In particular, for strong-Baxter permutations, the limiting skew Brownian permuton has parameters $|\rho|<1$ and $q\neq1/2$.

The conjecture seeks a natural permutation model realizing the genuinely two-parameter regime not covered by the paper's existing evidence, which has either $\rho=1$ or $q=1/2$.