We recently introduced a new universal family of permutons, depending on two parameters, called skew Brownian permuton. For some specific choices of the parameters, the skew Brownian permuton coincides with some previously studied permutons: the biased Brownian separable permuton and the Baxter permuton. The latter two permutons are degenerate cases of the skew Brownian permuton.
In the present paper we prove the first convergence result towards a non-degenerate skew Brownian permuton. Specifically, we prove that strong-Baxter permutations converge in the permuton sense to the skew Brownian permuton for a non-degenerate choice of the two parameters. In order to do that, we develop a robust technique to prove convergence towards the skew Brownian permuton for various families of random constrained permutations. This technique relies on generating trees for permutations, allowing an encoding of permutations with multi-dimensional walks in cones. We apply this technique also to semi-Baxter permutations.