The centralizer fusion-system conjecture for normal pairs

Let $((S,\mathcal{F}),(T,\mathcal{E}))$ be a normal pair of fusion systems realized by a normal pair $(G,H)$ of finite groups. Write $O_{p'}(H)$ for the largest normal subgroup of $H$ of order prime to $p$. Let $C_S(H)$ be the centralizer of $H$ in $S$, $C_G(H)$ the centralizer of $H$ in $G$, and $\mathcal{F}_{C_S(H)}(C_G(H))$ the fusion system induced by $C_G(H)$ on $C_S(H)$. Centralizer fusion-system conjecture. If $O_{p'}(H)=1$, then

$$C_\mathcal{F}(\mathcal{E})=\mathcal{F}_{C_S(H)}(C_G(H)).$$

This strengthens the preceding proposed equality by identifying the entire centralizer fusion system, rather than only its underlying centralizer subgroup. The source gives no resolution evidence, so the conjecture remains open.