Croke–Dairbekov–Sharafutdinov marked-length-spectrum volume conjecture

Let $N$ be a manifold with two negatively curved Riemannian metrics $\mathbf{g}_0$ and $\mathbf{g}_1$. For each free homotopy class $\langle\gamma\rangle$, let $\operatorname{MLS}_{\mathbf{g}}(\langle\gamma\rangle)$ be the length of its shortest representative. Write $\operatorname{MLS}_{\mathbf{g}_1}\geq\operatorname{MLS}_{\mathbf{g}_0}$ when this inequality holds in every free homotopy class.

Croke–Dairbekov–Sharafutdinov marked-length-spectrum volume conjecture. The inequality

$$\operatorname{MLS}_{\mathbf{g}_1}\geq\operatorname{MLS}_{\mathbf{g}_0}$$

should imply

$$\operatorname{vol}(\mathbf{g}_1)\geq\operatorname{vol}(\mathbf{g}_0).$$

Moreover, equality of volumes should hold if and only if $\mathbf{g}_0$ and $\mathbf{g}_1$ are isometric. This is a rigidity version of marked-length-spectrum volume comparison in negative curvature; the source gives no resolution.