Sharpness problem for the subgroup spectral sequence of a p-local finite group

Let $(S, {\mathcal F}, {\mathcal L})$ be a $p$-local finite group. The groups $E_2^{p,q}$ are the higher limits of the functor assigning to each object its mod-$p$ cohomology:

$$E_2^{p,q}=\underset{{\mathcal O}^c({\mathcal F})}{\lim{}^p}\,H^q(-;{\mathbb F}_p).$$

Sharpness problem. For every $p>0$ and $q\geq 0$,

$$E_2^{p,q}=\underset{{\mathcal O}^c({\mathcal F})}{\lim{}^p}\,H^q(-;{\mathbb F}_p)=0.$$

This vanishing would make the associated spectral sequence sharp. For fusion systems realized by a finite group, the statement is proved by Diaz and Park; the general case for $p$-local finite groups is the problem posed here.