Product decomposition conjecture for higher-rank group manifolds

Let $G_0$ be a real linear simple Lie group of real rank at least $2$, let $K$ be a maximal compact subgroup of $G_0$, and let $\Gamma$ be a discrete subgroup of $G_0\times G_0$ acting properly discontinuously and cocompactly on $(G_0\times G_0)/\operatorname{Diag}(G_0)$. Product decomposition conjecture. After replacing $\Gamma$ by a finite-index subgroup, one has $\Gamma=\Gamma_1\times\Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are discrete subgroups of $G_0$ with disjoint limit cones and

$$\operatorname{vcd}(\Gamma_1)+\operatorname{vcd}(\Gamma_2)=\dim(G_0/K).$$

This conjecture seeks a general classification of compact quotients of higher-rank group manifolds; the paper proves the analogous product-rigidity statement for restricted root systems of type $A_2$, while the general case remains open.