Geck's unitriangularity conjecture for unipotent modules

Let $cell$ be an odd prime, let $cell$-modular decomposition define the map $d_{\mathscr{U}}:[\mathscr{U}_K]\to[\mathscr{U}_{\operatorname{k}}]$, and let $E_\Theta$ denote the irreducible unipotent characteristic-zero module indexed by an odd-defect symbol $\Theta$. A simple unipotent module is denoted by $D_\Theta$, and $a(\Theta)$ is the $a$-function of the symbol.

Geck's conjecture. Assume $\ell$ is odd. There exists a parametrization

$${\operatorname{Irr}}(\mathscr{U}_{\operatorname{k}})=\{[D_\Theta]\mid \Theta\in\mathscr{S}_{\mathrm{odd}}\}$$

of the simple unipotent modules and an ordering $\leqslant$ on symbols such that

$$d_{\mathscr{U}}([E_\Theta])\in [D_\Theta]+\sum_{\Theta'<\Theta}\mathbb{Z}[D_{\Theta'}].$$

Moreover, the ordering $\leqslant$ can be chosen to be compatible with the $a$-function, so that $\Theta'\leqslant\Theta$ implies $a(\Theta')\leqslant a(\Theta)$.

This conjecture would provide a natural parametrization of the simple unipotent modules and a unitriangular decomposition matrix. It is known for linear primes, when the order of $q$ modulo $\ell$ is odd, and in various small-rank cases; the general unitary-prime case remains open.