Semisimplicity conjecture for the graded pieces $gr^N_jW$

Let $G$ be the group in the paper, let ${\mathcal I}_G$ be the indicated category of $G$-modules, and let $N$ denote the filtration used to define $gr^N_jW$. Semisimplicity conjecture. If $n=\infty$, then for any $j\ge 0$ and any object $W$ of ${\mathcal I}_G$, the $G$-module $gr^N_jW$ is semi-simple. Such semisimplicity would give a strong structural description of the filtered representation category; the source gives no evidence that this conjecture has been resolved.