Block orthogonality conjecture for principal blocks of \mathbb{Z}_\ell-spetses

Let $\mathbb{G}=(W,\Lambda)$ be a simply connected $\mathbb{Z}_\ell$-spets such that $\ell>2$ is very good for $\mathbb{G}$ and coprime with $|W|$. Let $q\in1+\ell\mathbb{Z}_\ell$, let $S$ be the Sylow $\ell$-subgroup of $\mathbb{G}(q)$, and let $B_0$ be the principal block of $\mathbb{G}(q)$. For $t\in S$, write $B_0(t)$ for the principal block of $C_{\mathbb{G}}(t)$, and write $t\sim_{\mathcal F}t'$ for $\mathcal F$-conjugacy. Block orthogonality conjecture. For all $t,t'\in S$,

$$\sum_{\chi \in \operatorname{Irr}(B_0)} \chi(t)\overline{\chi(t')} = \begin{cases} \dim(B_0(t))& \text{if $t\sim_{\mathcal F} t'$,}\\ 0& \text{otherwise.}\end{cases}$$

This is the spetsial analogue of block orthogonality for principal blocks of finite groups. It follows when the associated Hecke algebra is strongly symmetric, and the general assertion remains conjectural.