Shareshian's subgroup-lattice homotopy conjecture

Let $G$ be a finite group, and consider an open interval in the subgroup lattice of $G$. Shareshian's conjecture. Every such open interval has the same homotopy type as a wedge of spheres. The conjecture concerns the topology of subgroup lattices and is raised here in connection with observed inequalities involving maximal-dimensional intersections; no resolution is given in the supplied text.