Local rigidity conjecture for weakly irreducible translation actions

Let $G=G_1\times\cdots\times G_k$ be a semisimple Lie group with all factors non-compact, and let $\Lambda<H$ be a cocompact lattice, with $G<H$. A left translation action $\rho$ of $G$ on $H/\Lambda$ is weakly irreducible when every ergodic component for the action of any $G_i$ is an ergodic component for the action of $G$. Weak irreducibility conjecture. If $\rho$ is weakly irreducible, then $\rho$ is locally rigid. This is presented as the $G$-action variant of the preceding conjecture; the source gives no resolution.