Zelditch's nodal-measure equidistribution conjecture

Let $(M,g)$ be a compact smooth Riemannian manifold with ergodic geodesic flow, and let $(\psi_j)_{j\geq1}$ be any orthonormal basis of eigenfunctions. For an eigenfunction, let $\mu^Z_{\psi_j}$ denote the normalized probability measure induced by its nodal set. Zelditch's conjecture. The measures $\mu^Z_{\psi_j}$ weak-* converge to Lebesgue measure on $M$ as $j\to\infty$. This is presented as a spatial, nodal-set analogue of QUE; the supplied source gives no resolution.