Monotonicity conjecture for the random billiard walk variance

Let $b\in S^{d-1}$ be a direction, let $p\in(0,1)$ be the parameter, and let $\sigma_{b,p}^2$ denote the variance of the random billiard walk in direction $b$. Monotonicity conjecture. For every fixed $b\in S^{d-1}$, $\sigma_{b,p}^2$ is monotone in $p$. Little is known about this variance; the paper notes that it extends continuously to $S^{d-1}\times(0,1)$ and has a simple form for certain Coxeter directions, but does not provide a resolution of this monotonicity claim.