Gromov's vanishing conjecture for Lp-cohomology of symmetric spaces

Let $G$ be a semisimple Lie group, let $K$ be a maximal compact subgroup, and let $G/K$ be the associated homogeneous space. Write $\operatorname{rank}_{\mathbb{R}}(G)$ for the real rank of $G$. Gromov's conjecture. For $1<p<\infty$ and $k<\operatorname{rank}_{\mathbb{R}}(G)$,

$$H_p^k(G/K)=0.$$

The conjecture concerns vanishing of $L_p$-cohomology in degrees below the real rank of a semisimple Lie group. The source later considers this conjecture for symmetric spaces and states that it resolves part of it; the supplied text does not establish the full conjecture's status.