Schur critical-index conjecture for simple Lie groups and lattices

Let $\mathrm{G}$ be a simple Lie group or lattice, and define its Schur critical index by

$$q_{\mathrm{G}}:=\sup\big\{p\geq 2\colon \mathrm{G}\text{ is Schur weakly $p$-amenable}\big\}.$$

Schur critical-index conjecture. The critical index satisfies

$$q_{\mathrm{G}}=\frac{2\,\operatorname{rank}(\mathrm{G})}{\operatorname{rank}(\mathrm{G})-1}.$$

The source says this holds for $S L_2(\mathbf{R})$ and matches known results for $S L_3(\mathbf{R})$, while proposing the formula for simple Lie groups and lattices generally. Its Fourier analogue is described as potentially implying that the rank is retained by the group von Neumann algebra.