Weak Injective Radius Property implies the General Density and Spherical Density Hypotheses

Let a sequence of congruence subgroups be equipped with the Weak Injective Radius Property, the General Density Hypothesis, and the Spherical Density Hypothesis as defined in the source. Weak Injective Radius conjecture. The Weak Injective Radius Property implies both the General Density Hypothesis and the Spherical Density Hypothesis. This is stated as a belief in the discussion of relations among the properties. The source notes that the implication is already established in some settings, including the $p$-adic case for the relevant results, while the general Archimedean situation remains unresolved.