Local rigidity conjecture for irreducible affine actions without rank-one factors

Let $G=G_1\times\cdots\times G_k$ be a semisimple Lie group whose factors are all non-compact, let $\Gamma<G$ be an irreducible lattice, and let $\rho$ be an affine action of $\Gamma$ on $H/\Lambda$, where $H$ is a Lie group and $\Lambda<H$ is a cocompact lattice. Say that $\rho$ has rank-one factors if it has a factor action obtained by projecting $\Gamma$ to a rank-one factor $G_i$ and restricting a $G_i$ action. Irreducible affine-action conjecture. If $\rho$ has no rank-one factors, then $\rho$ is locally rigid. The conjecture is motivated by higher-rank rigidity and excludes the rank-one-factor examples; the source gives no resolution.