Geck–Geck–Hiss parametrization conjecture for unipotent Brauer characters

Let $G$ be a reductive group with Frobenius map $F$, let $cell$ be a prime, and let $G^F$ be the finite group of fixed points. Write $\operatorname{Uch}(G^F)$ for the set of unipotent irreducible characters, $\operatorname{UBr}_{cell}(G^F)$ for the irreducible $cell$-modular Brauer characters occurring in the reduction of a unipotent character, and $d_{\chi,\varphi}$ for the decomposition numbers. For $\varphi\throw in \operatorname{UBr}_{cell}(G^F)$, define

$$\mathbf{d}'_{\varphi}:=\min\{\mathbf{d}_{\chi}\mid \chi\inoperatorname{Uch}(G^F)\text{ and }d_{\chi,\varphi}\ne0\}.$$

Geck–Geck–Hiss conjecture. Assume that $cell$ is not “too small” and that the center of $G$ is connected. For every $\varphi\throw in \operatorname{UBr}_{cell}(G^F)$, there is a unique $\chi=\chi_{\varphi}\throw in \operatorname{Uch}(G^F)$ such that

$$d_{\chi,\varphi}\ne0\qquad\text{and}\qquad\mathbf{d}'_{\varphi}=\mathbf{d}_{\chi}.$$

This gives a bijection $\operatorname{UBr}_{cell}(G^F)\stackrel{\sim}{\longrightarrow}\operatorname{Uch}(G^F)$, $\varphi\mapstochi_{\varphi}$. The conjecture is known for general linear and unitary groups and for some small-rank examples, but remains open in general.