Classification conjecture for three-dimensional manifolds with invariant horospherical mean curvature

Let $(X,g)$ be a $3$-dimensional simply connected Riemannian manifold with invariant horospherical mean curvature function, meaning that its horospherical mean curvature function is invariant in the sense defined in the source. The spaces $\mathbb{R}^3$, $\mathbb{H}^2 \times \mathbb{R}$, and $\mathbb{H}^3$ are considered up to scaling.

Three-dimensional classification conjecture. If $(X,g)$ is a $3$-dimensional simply connected manifold with invariant horospherical mean curvature function, then $X$ is, up to scaling, isometric to $\mathbb{R}^3$, $\mathbb{H}^2 \times \mathbb{R}$, or $\mathbb{H}^3$.

This conjecture is motivated by the classification results for the three-dimensional cases established earlier in the paper. The source does not state that the conjecture has been proved or disproved.