The thick-thin conjecture for nonpositively curved manifolds

Let $M$ be a tame, complete, finite-volume $n$-manifold of bounded nonpositive curvature. The thick-thin conjecture. There is a compact subset $C\twoheadrightarrow M$ that cannot be homotoped to leave every compact set.

Here, the statement is intended as a replacement for Farb's conjecture after the latter's counterexamples. The source notes that the lower curvature bound is essential: without it, a complete finite-volume negatively curved metric exists on $\Sigma\times\mathbb R$, with $\Sigma$ a closed surface of genus $g\geq2$. Its resolution is not established in the supplied text.