Craven's perversity-unitriangularity conjecture for unipotent blocks

Let $G={\mathbf{G}}^F$ for ${\mathbf{G}}$ connected reductive defined over ${\mathbb{F}}_q$. Let $\ell$ be a prime such that Sylow $\ell$-subgroups of $G$ are abelian, and let $d=d_\ell(q)$ be the order of $q$ in ${\mathbb{F}}_\ell^\times$. For each unipotent character $\rho$ of $G$, let $\Psi_{\vhi_\rho}$ be the corresponding irreducible Brauer character, and let $\pi_d(\rho)$ be Craven's perversity function. Then, for any two unipotent characters $\rho\ne\rho'$ of $G$, Craven's conjecture.

$$\langle \rho,\Psi_{\vhi_{\rho'}} \rangle \neq 0 \Longrightarrow \pi_d(\rho)>\pi_d(\rho').$$

This is the unitriangularity predicted by Craven's conjectural perverse equivalences between unipotent $\ell$-blocks and their Brauer correspondents. It refines the ordering by the $a$- and $A$-functions, but the supplied text does not establish the assertion or provide evidence resolving it.