The projective-degree divisibility conjecture for principal blocks of spetses

Let $\mathbb{G}=(W\varphi^{-1},L)$ be as in the principal-block dimension conjecture, and suppose that $W$ is an $\ell'$-group, $\varphi$ is trivial, and $q\equiv1\pmod\ell$. Let $\operatorname{IBr}(B_0)$ denote the proposed analogue of the irreducible Brauer characters of $B_0$, and let $\deg\Phi_\nu$ denote the corresponding formal degree of a projective indecomposable character. Projective-degree divisibility conjecture. For every $\nu\in\operatorname{IBr}(B_0)$,

$$|S|\mid\deg\Phi_\nu|_{x=q}.$$

The paper proves this in several special cases and restates it through a $W$-equivariant bijection with characters of $SW$; the general assertion is not established.