The measurable-to-smooth invariant-metric conjecture

Let $G$ be a semisimple Lie group with property $(T)$, or let $\Gamma$ be a lattice in such a group, acting smoothly on a compact manifold $M$ as in the source. The measurable-to-smooth invariant-metric conjecture. If the acting group preserves a measurable Riemannian metric, then it should preserve a smooth invariant Riemannian metric. This is a technical conjecture intended to promote measurable rigidity to smooth rigidity and remains open.