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We recently introduced a new universal family of permutons, depending on twoparameters, called skew Brownian permuton. For some specific choices of theparameters, the skew Brownian permuton coincides with some previously studiedpermutons: the biased Brownian separable permuton and the Baxter permuton. Thelatter two permutons are degenerate cases of the skew Brownian permuton.  In the present paper we prove the first convergence result towards anon-degenerate skew Brownian permuton. Specifically, we prove thatstrong-Baxter permutations converge in the permuton sense to the skew Brownianpermuton for a non-degenerate choice of the two parameters. In order to dothat, we develop a robust technique to prove convergence towards the skewBrownian permuton for various families of random constrained permutations. Thistechnique relies on generating trees for permutations, allowing an encoding ofpermutations with multi-dimensional walks in cones. We apply this techniquealso to semi-Baxter permutations.

The permuton limit of strong-Baxter and semi-Baxter permutations is the skew Brownian permuton