Spatzier's rank-rigidity lattice conjecture

Let $G$ be a simple Lie group with real rank at least $2$. For a countable group $\Gamma$, let $A_i$ consist of elements whose centralizer contains a free abelian subgroup of rank at most $i$ as a finite-index subgroup, and define the rank of $\Gamma$ to be the least $i$ for which

$$\Gamma=\gamma_1A_i\cup\cdots\cup\gamma_mA_i$$

for some finite set $\gamma_1,\ldots,\gamma_m\in\Gamma$. Let $\Gamma$ be a Zariski-dense discrete subgroup of $G$ whose rank equals the real rank of $G$. Spatzier's conjecture. Then $\Gamma$ is a lattice in $G$. This conjecture is inspired by rank rigidity in differential geometry; the source also notes a related question asking whether rank at least $2$ suffices.