Homotopy complexity conjecture for arithmetic manifolds

Let $X$ be a symmetric space of non-compact type. For positive real numbers $\delta$ and $\alpha$, a $(\delta,\alpha\operatorname{vol}(M))$-simplicial complex is understood in the sense used for simplicial complexity in the source, and let $M$ be an arithmetic $X$-manifold.

Homotopy complexity conjecture. There are $\alpha>0$ and $\delta>0$ such that every arithmetic $X$-manifold $M$ is homotopy equivalent to a $(\delta,\alpha\operatorname{vol}(M))$-simplicial complex. This conjecture predicts a linear-in-volume bound on the simplicial complexity of arithmetic locally symmetric manifolds. It is recalled in the discussion of extending results from negatively curved to non-positively curved manifolds; the source gives no resolution status.