Hamiltonian volume conjecture for vacuum data on hyperbolic 3-manifolds

Let $M$ be a compact $3$-dimensional manifold of hyperbolic type. Let $(g,K)$ be vacuum data on $M$, meaning that the vacuum Einstein constraint equations hold, and suppose that the mean curvature $\operatorname{tr}_g K=\tau$ is constant. Define the reduced Hamiltonian by

$$\mathcal H(g,K)=|\operatorname{tr}_g K|^3\operatorname{Vol}(M,g).$$

Hamiltonian volume conjecture. The rescaled volume satisfies

$$\mathcal H(g,K)\geq 3^3\operatorname{Vol}(M,g_0),$$

where $g_0$ is the hyperbolic metric on $M$ with sectional curvature $-1$. Equality holds if and only if $g$ is isometric to $\frac{n^2}{\tau^2}g_0$ and $K=\frac{\operatorname{tr}K}{n}g$. This is proposed as an analogue of the positive mass theorem for spatially compact spacetimes and is presented as a consequence of the sigma-constant conjecture; the source does not specify whether it has been resolved.