Benson's generation conjecture for cochains on compact Lie groups

Let $G$ be a compact Lie group with a normal maximal torus $T$, let $C_*G$ be the chain DG algebra of $G$, and let $C^*BG$ be its cochain algebra. Write $\mathcal{D}^b(C^*BG)$ for the bounded derived category and $\mathcal{D}(C^*BG)$ for the derived category.

Benson's generation conjecture. The category $\mathcal{D}^b(C^*BG)$ is generated as a thick subcategory of $\mathcal{D}(C^*BG)$ by the objects $\operatorname{Hom}_{C_*G}(k,X)$ for $X$ in $\mathcal{D}^b(C_*G)$.

This is one of two generation conjectures transported from finite groups to compact Lie groups. The supplied text does not state whether it has been proved or disproved.