Reflexivity characterization by quasihyperbolic geodesics in open half spaces

Let $X$ be a strictly convex Banach space, and equip each open half space of $X$ with its quasihyperbolic metric. A domain is geodesic if every pair of its points can be joined by a quasihyperbolic geodesic.

Reflexivity conjecture. The Banach space $X$ is reflexive if and only if each of its open half spaces is geodesic in the quasihyperbolic metric.

The only-if direction is established in the source, since reflexivity implies the Radon–Nikodym property. The conjectural part is the converse: if every open half space is quasihyperbolically geodesic, then $X$ must be reflexive.