Margulis' short-geodesic counting conjecture for arithmetic lattices

Let $G$ be the ambient semisimple Lie group, let $X$ be its symmetric space, and let $\Gamma$ range over arithmetic lattices with $\operatorname{vol}(\Gamma\backslash X)\leq V$. For $\gamma\in\Gamma$, define the displacement function $d_\gamma(x)=d(\gamma x,x)$, and let $\{\gamma\}_\Gamma$ denote its conjugacy class in $\Gamma$. Margulis' conjecture. There exist positive constants $\varepsilon$ and $\alpha$ such that

$$\sup_{\Gamma\,:\, \operatorname{vol}(\Gamma\backslash X)\leq V}\#\{\{\gamma\}_\Gamma: \gamma\in\Gamma,\ \min d_\gamma\leq\varepsilon\}\ll V^{1-\alpha}.$$

The conjecture would provide sublinear control of short closed geodesics and, according to the source, would remove an injectivity-radius hypothesis from a preceding result.