Weak uniform discreteness conjecture for positive-characteristic simple groups

Let $k$ be a local field of positive characteristic, let ${\mathbb G}$ be a simply connected absolutely almost simple $k$-group with positive $k$-rank, and let $G={\mathbb G}(k)$ be the group of $k$-rational points. A space of discrete invariant random subgroups is weakly uniformly discrete if, for every $\epsilon>0$, there is an identity neighbourhood $U\subset G$ such that every $\mu\in\operatorname{DIRS}(G)$ satisfies

$$\mu\bigl(\{\Gamma\in\operatorname{Sub}(G):\Gamma\cap U=\{1\}\}\bigr)>1-\epsilon.$$

Weak uniform discreteness conjecture. The space $\operatorname{DIRS}(G)$ is weakly uniformly discrete. The statement extends weak uniform discreteness beyond the no-discrete-small-subgroups setting discussed earlier in the paper; the source provides no resolution.