Attracting-face convergence conjecture for degenerate diffusions

Let $\mathbb{X}^{\circ}$ denote the interior of the state cube, let $\mathbb{F}$ be the collection of faces, and let $F\in\mathbb{F}$ be an attracting face supporting the invariant probability measure $\pi_F$. For an initial point $x\in\mathbb{X}^{\circ}$, write $X^x_\omega(t)$ for the diffusion path, $\bar F$ for the closure of $F$, and $\mu^x_{\omega,t}$ for its empirical measure. Attracting-face convergence conjecture. Under broad conditions guaranteeing that there is at most one invariant measure concentrated on each face, for each attracting $F\in\mathbb{F}$ and every $x\in\mathbb{X}^{\circ}$, with positive probability, $\bar F$ is the set of limit points of $X^x_\omega(t)$ as $t\to\infty$ and the empirical measures $\mu^x_{\omega,t}$ converge to $\pi_F$ as $t\to\infty$. This asserts positive-probability convergence of the diffusion to each attracting face together with convergence of its empirical statistics to the face-supported invariant measure. The source does not provide a resolution of the claim, and the breadth of the stated conditions remains unspecified.