Green [5] conjectured that if $M$ is a closed Riemannianmanifold of negative sectional curvature such that the meancurvatures of the horospheres through each point depend only onthe point, then $V$ is a locally symmetric space of rank one. Heproved this in dimension two. In this paper we prove that underGreen's assumption, $M$ must be asymptotically harmonic andthat the geodesic flow on $M$ is $C^{\infty}$ conjugate to that ofa locally symmetric space of rank one. Combining this with therecent rigidity theorem of Besson–Courtois–Gallot [1], itfollows that Green's conjecture is true for all dimensions.