Monotonicity of inversion probability in skew Brownian permutons

Let $\bm \mu_{\rho,q}$ be a skew Brownian permuton with parameters $(\rho,q)\in(-1,1]\times[0,1]$, and define

$$f(\rho,q)=\mathbb{P}(\operatorname{Perm}_2(\bm \mu_{\rho,q})=21).$$

Here $21$ is the inversion pattern of size two. Inversion monotonicity conjecture. For every fixed $\rho\in(-1,1]$, the function $f(\rho,q)$ is increasing in $q$. An explicit formula for $f(\rho,q)$ is posed as a further problem, and the supplied text does not give a resolution of this monotonicity claim.