The principal-block dimension conjecture for -spetses

Let $\ell>0$ be prime. Let $\mathbb{G}=(W\varphi^{-1},L)$ be a simply connected $\mathbb{Z}_\ell$-spets for which $\ell$ is very good, let $q$ be a power of a prime different from $\ell$, and let $B_0$ be the principal $\ell$-block of $\mathbb{G}(q)$ with defect group $S$. Define

$$\dim(B_0):=\sum_{\gamma\in\operatorname{Irr}(B_0)}\gamma(1)^2\in\mathbb{Z}[x].$$

Principal-block dimension conjecture.

$$\left(\dim(B_0)|_{x=q}\right)_\ell=|S|$$

and

$$\left(\dim(B_0)|_{x=q}\right)_{\ell'}\equiv |W_{\varphi^{-1}\zeta^{-1}}|_{\ell'}\pmod\ell,$$

where $\zeta\in\mathbb{Z}_\ell^\times$ satisfies $q\equiv\zeta\pmod\ell$, and $W_{\varphi^{-1}\zeta^{-1}}$ is the associated relative Weyl group. This combines the two stated dimension and global-local assertions; the paper proves it under additional hypotheses, but the general case is described as elusive.