  Let $G$ be a non-abelian finite simple group. In addition, let $\Delta_G$ be
the intersection graph of $G$, whose vertices are the proper nontrivial
subgroups of $G$, with distinct subgroups joined by an edge if and only if they
intersect nontrivially. We prove that the diameter of $\Delta_G$ has a tight
upper bound of 5, thereby resolving a question posed by Shen (2010).
Furthermore, a diameter of 5 is achieved only by the baby monster group and
certain unitary groups of odd prime dimension.
