Green--Wasserstein Inequality on Compact Surfaces

Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary. In this note, we answer a question posed by Steinerberger: can one remove the $\sqrt{\log n}$ factor in the two-dimensional Green--Wasserstein inequality while keeping the unrenormalized off-diagonal Green term? We show that this is impossible on any compact connected surface: there is no inequality of the same form that holds uniformly over point sets with an $O(n^{-1/2})$ remainder for all $n$. We argue by contradiction and combine a second-moment estimate for the random Green energy of i.i.d. samples with the semi-discrete random matching asymptotics of Ambrosio--Glaudo.