The property-(T) invariant-metric conjecture

Let $\Gamma$ be a group with property $(T)$ acting smoothly on a compact manifold $M$ and preserving volume. Let $d(\Gamma)$ be the smallest dimension in which $\Gamma$ has an infinite-image linear representation. The invariant-metric conjecture. If $\dim(M)<d(\Gamma)$, the action should preserve a smooth Riemannian metric. Measurable invariant metrics are known in several settings, but smoothness remains the conjectural step.