Equivalences between fusion systems of finite groups of Lie type

We prove, for certain pairs $G,G'$ of finite groups of Lie type, that the $p$-fusion systems $\mathcal{F}_p(G)$ and $\mathcal{F}_p(G')$ are equivalent. In other words, there is an isomorphism between a Sylow $p$-subgroup of $G$ and one of $G'$ which preserves $p$-fusion. This occurs, for example, when $G=\mathbb{G}(q)$ and $G'=\mathbb{G}(q')$ for a simple Lie ``type'' $\mathbb{G}$, and $q$ and $q'$ are prime powers, both prime to $p$, which generate the same closed subgroup of $p$-adic units. Our proof uses homotopy-theoretic properties of the $p$-completed classifying spaces of $G$ and $G'$, and we know of no purely algebraic proof of this result.