  The Laplace-Beltrami eigenfunctions on a compact Riemannian manifold $M$
whose geodesic billiard flow has mixed character have been conjectured by
Percival to split into two complementary families, with all semiclassical mass
supported in the completely integrable and ergodic regions of phase space
respectively. In this paper, we consider the Dirichlet Laplacian on a family of
mushroom billiards $M_t$ parametrised by the length $t\in(0,2]$ of their
rectangular part. We prove that $M_t$ satisfies Percival's conjecture for
almost all $t\in(0,2]$, hence providing the first example of a billiard known
to satisfy Percival's conjecture.
