The well-definedness conjecture for skew Brownian permutons

For $\rho\in[-1,1]$ and $q\in[0,1]$, let $\bm\mu_{\rho,q}$ denote the skew Brownian permuton, defined from the solution flow through the push-forward of Lebesgue measure on $[0,1]$.

Skew Brownian permuton well-definedness conjecture. The skew Brownian permuton is well-defined for every parameter pair

$$(\rho,q)\in[-1,1]\times[0,1].$$

This is stated as a corollary of the preceding conjectures concerning the biased-sign construction and existence and pathwise uniqueness of the SDE flow.