Define a point cloud $X=\{x_i\}_{1\leq i\leq n}$, for $x_i\in\mathbb R^d$. Define the Wasserstein kernel as $$W(X,Y)=\max_{T}\frac{1}{n}\sum_{kl}T_{kl}\langle x_k,y_l \rangle$$ where $T$ is any doubly stochastic matrix. Consider point clouds $X_1,...X_m$ each of size $n$ and their Gram matrix $(W(X_i,X_j))_{ij}$. Is it positive semi-definite?

What we know: There always exists a permutation matrix $T$ maximizing the above.