Asymptotically harmonic analogue of Lichnerowicz's conjecture

Let $(M,g)$ be a complete, simply connected Riemannian manifold without conjugate points. The manifold $M$ is asymptotically harmonic if the mean curvature of all its horospheres is the same nonnegative constant.

Asymptotically harmonic classification conjecture. If $M$ is an asymptotically harmonic manifold, then $M$ is either flat or a rank-one symmetric space of noncompact type.

This conjecture is presented as an analogue of Lichnerowicz's conjecture for asymptotically harmonic manifolds. The source says that it had been partially resolved, citing several results, but does not establish a complete resolution.