Generic density conjecture for wave fronts on Riemannian manifolds

Let $M$ be a $d$-dimensional Riemannian manifold with or without boundary, where $d\geq 2$, and let $W_t(P)$ denote the wave front at time $t$ from a point $P\in M$. The wave front becomes dense on $M$ if it eventually intersects every open metric ball in $M$. Generic wave-front density conjecture. For a generic such manifold, the wave front $W_t(P)$ becomes dense for all points $P\in M$. This is motivated by numerical experiments and the expected growth of the wave front, but the meaning of genericity is only discussed informally and the conjecture remains open.