  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 $p = 2$.
