The intersection graph of a finite simple group has diameter at most 5

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.