Conjecture on asymptotically harmonic manifolds

Let $(M,g)$ be a complete, simply connected Riemannian manifold without conjugate points. The manifold $(M,g)$ is asymptotically harmonic if there is a non-negative constant $h$ such that every Busemann function $b_v$ satisfies $\Delta b_v\equiv h$.

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

This conjecture extends Lichnerowicz's conjecture from harmonic to asymptotically harmonic manifolds. The claim is known in dimension $3$, while the general classification remains open.