Wave-front density conjecture for smooth metrics on tori

Let $\mathbb{T}^d$ be the $d$-dimensional torus with $d>1$, equipped with a smooth Riemannian metric, and let $W_t(P)$ denote the wave front at time $t$ from a point $P\in\mathbb{T}^d$. The wave front becomes dense if it eventually intersects every open metric ball in the torus. Torus wave-front density conjecture. For any smooth Riemannian metric on $\mathbb{T}^d$ with $d>1$, the wave front of any point $P$ becomes dense. The claim extends the proved flat-torus case to arbitrary smooth metrics; the source states that this extension has not been proved, although hyperbolic geodesic flow would imply density.