Sharpness for fusion systems

Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$. Write $\mathcal{O}(\mathcal{F}^c)$ for the centric orbit category, whose objects are the $\mathcal{F}$-centric subgroups of $S$, and consider, for each $j\geq 0$, the cohomology functor with trivial coefficients

$$H^j(-;\mathbb{F}_p)\colon\mathcal{O}(\mathcal{F}^c)\rightarrow \mathbb{F}_p\text{-}\mathsf{mod}.$$

Sharpness for fusion systems. For all $i\geq 1$ and $j\geq 0$, the higher limits vanish:

$$\varprojlim^i_{\mathcal{O}(\mathcal{F}^c)} H^j(-;\mathbb{F}_p)=0.$$

This asserts that the subgroup decomposition of the classifying space of a saturated fusion system is sharp, generalizing the corresponding result for finite groups proved by Dwyer. The conjecture remains open in the general fusion-system setting.