The irreducible product-action conjecture

Let $\Gamma=\Gamma_1\times\Gamma_2$, where an action is irreducible when each factor acts ergodically on the underlying measure space. Assume that $\Gamma_1$ has property $(T)$ and no unbounded linear representations, and that $\Gamma_2$ is amenable. The irreducible product-action conjecture. There should be no irreducible, volume-preserving action of $\Gamma$ on a compact manifold. This extends known results for products involving solvable factors and remains open.