Scalar-curvature volume-entropy conjecture for closed 3-manifolds

Let $(M,g)$ be a closed Riemannian $3$-manifold, let $R(g)$ denote its scalar curvature, and let $h(M,g)$ denote the volume entropy of $g$. Let $h(\mathbb{H}^{3})$ denote the volume entropy of hyperbolic $3$-space.

Volume-entropy conjecture. If

$$R(g)\geq -6,$$

then

$$h(M,g)\leq h(\mathbb{H}^{3}).$$

The source says this would provide another proof of the main theorem using the natural-map methods, but gives no proof or resolution.