Drift classification conjecture for Markov-modulated diffusions

Let $\mathcal{A}$ be a manifold in $\mathbb{R}^d$ without boundary, and let $(X_t,\eta_t)$ be a diffusion process on $\mathbb{R}\times\mathcal{A}$. Assume that $\eta_t$ has stationary distribution $\pi$, that $\mu:\mathcal{A}\to\mathbb{R}$ is bounded and continuous, and that the diffusion coefficient $\sigma$ is bounded away from zero and infinity. Drift classification conjecture. If

$$\int_{\mathcal{A}}\pi(\mathrm{d}x)\,\mu(x)>0,$$

then $X_t\to\infty$; if

$$\int_{\mathcal{A}}\pi(\mathrm{d}x)\,\mu(x)<0,$$

then $X_t\to-\infty$. This conjectured classification extends the corresponding discrete half-strip model to continuum Markov-modulated diffusions; the zero-drift case is not asserted here and remains untreated.