Invariance conjecture for the random refraction billiard walk

Consider the random refraction billiard walk $(R_t)_{t\geq0}$ in $\mathbb{R}^d$, started at $R_0\in\mathbb{R}^d$, with direction $b\in S^{d-1}$ and parameter $p\in(0,1)$. For $t\in[0,1]$, define the rescaled trajectories by

$$R_t^{(n)}=\frac{1}{\sqrt n}R_{tn}.$$

Invariance conjecture. For every $R_0\in\mathbb{R}^d$, $b\in S^{d-1}$ and $p\in(0,1)$, the rescaled trajectories $R_t^{(n)}$, $t\in[0,1]$, converge to isotropic Brownian motion in $\mathbb{R}^d$. Simulations and preliminary results are said to suggest behavior similar to that of the random reflection billiard walk, but the conjecture remains unresolved.