CSXDH-shape conjecture

Let $E\in\Pi^{\mathrm{alg}}(G)$ and let $\mathcal{M}(G)$ be the set of $A$-shapes. For $\varsigma\in\mathcal{M}(G)$, let $h(G,\varsigma;q,E)$ be the associated cohomological counting function and let $r(\varsigma)$ be its worst-case rate of decay of matrix coefficients. CSXDH-shape conjecture. Fix $E\in\Pi^{\mathrm{alg}}(G)$. Then for any $\varsigma\in\mathcal{M}(G)$,

$$h(G,\varsigma;q,E)\ll\operatorname{vol}(X(q))^{\frac{2}{r(\varsigma)}}.$$

The source describes this as a stronger version of the cohomological Sarnak–Xue density hypothesis; no resolution is given.