Dihedral rigidity conjecture for hyperbolic polyhedra

For a unit vector $cvec{a}=(a^1,a^2,a^3)$ and $s oindent\text{in}cmathbb{R}$, let

$$Z(cvec{a},s)=cbrace{x oindent\text{in}cmathbb{R}^3:x^1>0,csum_i a^i x^i=scbrace}$$

be the totally umbilic planes in the upper-half-space model $(cmathbb{R}^3_+,b)$ of hyperbolic $3$-space. Let $\bar P$ be a hyperbolic reference polyhedron enclosed by such planes and convex in $(\mathbb{R}^3_+,\delta)$, with faces $\bar F_i$. Let $(M^3,g)$ be a Riemannian manifold diffeomorphic to $\bar P$, and define the outward dihedral angle by $\cos\measuredangle_{ij}M=-\langle X_i,X_j\rangle$. Dihedral rigidity conjecture. If $R_g\geq-6$, the mean curvature of every face of $M$ is at least that of the corresponding face of $\bar P$, and every dihedral angle of $M$ is at most the corresponding angle of $\bar P$, then $M$ is isometric to $\bar P$. Mean curvature is computed with respect to the outward unit normal. This conjecture is motivated by hyperbolic mass and earlier dihedral rigidity conjectures; its resolution status is not established in the supplied text.