The volume-controlled homotopy type conjecture for locally symmetric manifolds

Let $S$ be a symmetric space of non-compact type, and let an $S$-manifold be a complete Riemannian manifold locally isometric to $S$. An irreducible $S$-manifold has no nontrivial locally symmetric product decomposition. A $(d,v)$-simplicial complex is a simplicial complex with at most $v$ vertices, all of them of valence at most $d$. The volume-controlled homotopy type conjecture. There are constants $\alpha(S),d(S)$ such that every irreducible $S$-manifold $M=\Gamma\backslash S$, assumed arithmetic when $\dim(S)=3$, is homotopy equivalent to a $\bigl(d(S),\alpha(S)\operatorname{vol}(M)\bigr)$-simplicial complex. This conjecture asserts that the topological complexity of locally symmetric manifolds is controlled linearly by volume; the paper proves it in many cases, while the stated general result is not resolved here.