Optimal Weak Injective Radius and General Density conjecture for Archimedean congruence subgroups

Let $G$ be an Archimedean, semisimple, almost-simple and simply connected group, let $\Gamma_1$ be an arithmetic lattice, and let $(\Gamma_N)$ be a sequence of congruence subgroups of $\Gamma_1$. Optimal Weak Injective Radius conjecture. The sequence $(\Gamma_N)$ satisfies the Weak Injective Radius Property with parameter $\alpha=1$; if $\Gamma_1$ is cocompact, it also satisfies the General Density Hypothesis with parameter $\alpha=1$. Consequently, $(\Gamma_N)$ has the Optimal Lifting Property. The conjecture seeks the optimal parameter in the Archimedean arithmetic setting. The source does not specify a resolution, although it records partial results and stronger consequences in related cases.