Linear homotopy complexity conjecture for locally symmetric spaces

Let $X$ be a symmetric space of non-compact type. Suppose that $\dim X\ne 3$. An irreducible complete Riemannian manifold $M$ locally isometric to $X$ is a manifold whose Riemannian volume is denoted by $\operatorname{vol}(M)$.

Linear homotopy complexity conjecture. There are constants $\alpha$ and $d$, depending only on $X$, such that every such $M$ is homotopically equivalent to a simplicial complex $\mathcal{R}$ with at most

$$\alpha\cdot\operatorname{vol}(M)$$

vertices, and all vertex degrees bounded by $d$.

This conjecture asserts that the homotopy complexity of locally symmetric spaces is bounded linearly by their volume. The theorem preceding it gives a weaker bounded-presentation statement; the context identifies the claim as conjectural for the exceptional cases other than $\operatorname{PSL}_2(\mathbb{C})$, while the general status is not resolved in the supplied text.