The homotopy type conjecture for arithmetic manifolds

Let $X$ be a symmetric space of non-compact type. An arithmetic $X$-manifold is an arithmetic manifold locally modeled on $X$, and a family has uniform homotopy complexity if there are constants $D,\alpha$ such that every manifold $M$ in the family is homotopy equivalent to a $(D,\alpha\cdot\operatorname{vol}(M))$-simplicial complex. Homotopy type conjecture. The family of arithmetic $X$-manifolds has uniform homotopy complexity. The paper proves only a weak variant of this conjecture for compact arithmetic manifolds; the full assertion is not resolved here.