Frobenius divisibility conjecture for partial character tables of spetses

Let $\mathbb{G}=(W,\Lambda)$ be a simply connected $\mathbb{Z}_\ell$-spets with $\ell>2$ and $(|W|,\ell)=1$. Let $T$ be the Sylow $\ell$-subgroup of $\mathbb{G}(q)$, let $\mathcal F$ be its fusion system, and let $\chi(t)$ denote the partial character-table value. Frobenius divisibility conjecture. For each $\chi\in\mathcal E_\ell(\mathbb{G}(q),1)$,

$$\left(\sum_{t\in T/W}|\mathbb{G}:C_{\mathbb{G}}(t)|\chi(t)\right)_{x=q}\equiv0\pmod{|T|}.$$

This is the spetsial analogue of Frobenius' theorem on character values. The conjecture holds when $W$ is rational, and the source defers a thorough investigation of the remaining cases.