Alperin's Weight Conjecture for connected \ell-local compact groups

Let $\mathcal F$ be an $\ell$-local compact group on a discrete $\ell$-toral group $S$ with maximal discrete torus $T$. Write $W=\operatorname{Out}_{\mathcal F}(T)$ for its Weyl group, and let $\mathbf w(\mathcal F)$ denote the number of weights of $\mathcal F$ up to conjugacy, where weights are defect-zero characters of $\mathcal F$-centric radical subgroups.

Alperin's Weight Conjecture. If $\mathcal F$ is connected, meaning every element of $S$ is $\mathcal F$-conjugate to an element of $T$, then

$$\mathbf w(\mathcal F)=|\operatorname{Irr}(W)|.$$

This extends Alperin's Weight Conjecture to connected $\ell$-local compact groups. The conjecture holds when the $\ell$-adic valuation of $|W|$ is zero, and the paper proves it when $\mathcal F$ is simple and $v_\ell(|W|)=1$; it remains open in general.