Gromov's positivity conjecture for negatively Ricci-curved manifolds

Let $M$ be a closed, connected, oriented $n$-dimensional manifold. Its simplicial volume is

$$\|M\| = \inf \left\{\sum_i |a_i|: \left[\sum_i a_i \sigma_i\right] = [M] \in H_n(M, \mathbb{R})\right\}.$$

A Riemannian metric on $M$ has negative definite Ricci curvature when its Ricci curvature is negative definite at every point. Gromov's positivity conjecture. If $M$ admits a Riemannian metric with nonpositive sectional curvature and negative definite Ricci curvature, then

$$\|M\| > 0.$$

Connell and Wang proved the claim in dimension three and established related positivity results under weaker curvature assumptions. The conjecture remains open in the generality stated.