The degree conjecture in nonpositive curvature

Let $M$ be a closed $n$-manifold with nonpositive sectional curvature, negative Ricci curvature, and no local $\mathbb{R}$ factors. Let $N$ be any closed Riemannian manifold and let $f:N\longrightarrow M$ be any continuous map. Degree conjecture in nonpositive curvature. There is a constant $C>0$, depending only on $n$ and the smallest Ricci curvatures of $N$ and $M$, such that

$$\operatorname{deg}(f)\leq C\frac{\operatorname{Vol}(N)}{\operatorname{Vol}(M)}.$$

The conjecture is known for negatively curved targets and, up to the stated exceptions, for locally symmetric targets by the degree theorem; the general nonpositively curved case remains open.