Maximal subgroups of exceptional groups and Quillen's dimension

Given a finite group $G$ and a prime $p$, let $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. The group $G$ satisfies the Quillen dimension property at $p$ if $\mathcal{A}_p(G)$ has non-zero homology in the maximal possible degree, which is the $p$-rank of $G$ minus $1$. For example, D. Quillen showed that solvable groups with trivial $p$-core satisfy this property, and later, M. Aschbacher and S.D. Smith provided a list of all $p$-extensions of simple groups that may fail this property if $p$ is odd. In particular, a group $G$ with this property satisfies Quillen's conjecture: $G$ has trivial $p$-core and the poset $\mathcal{A}_(G)$ is not contractible. In this article, we focus on the prime $p = 2$ and prove that the $2$-extensions of the exceptional finite simple groups of Lie type in odd characteristic satisfy the Quillen dimension property, with only finitely many exceptions. We achieve these conclusions by studying maximal subgroups and usually reducing the problem to the same question in small linear groups, where we establish this property via counting arguments. As a corollary, we reduce the list of possible components in a minimal counterexample to Quillen's conjecture at $p = 2$.