Existence of a dense wave front in every convex billiard

Let a convex billiard table be a convex planar domain, and let $W_t(P)$ denote the billiard wave front at time $t$ from a point $P$ in the table. The wave front becomes dense if it eventually intersects every open subset of the billiard table. Convex-billiard existence conjecture. For all convex billiard tables, there is a point $P$ such that the wave front $W_t(P)$ becomes dense. This is presented as a further plausible statement beyond generic density and is not known in the source.