A lack of Ricci bounds for the entropic measure on Wasserstein space over the interval

We examine the entropic measure, recently constructed by von Renesse and Sturm, a measure over the metric space of probability measures on the unit interval equipped with the 2-Wasserstein distance. We show that equipped with this measure, Wasserstein space over the interval does not admit generalized Ricci lower bounds in the entropic displacement convexity sense of Lott–Villani–Sturm. We discuss why this is contrary to what one might expect from heuristic considerations.