Let $(X,d)$ be a complete and separable metric space and, for $1\leq p<\infty$, let $(\mathcal{P}_p(X,d),W_p)$ be the $p$-Wasserstein space on $(X,d)$. For which $p$ and $(X,d)$ is $(\mathcal{P}_p(X,d),W_p)$ a $CAT(\kappa)$ space?

I know that for $p=2$ and $(X,d)$ a Banach space, $(\mathcal{P}_2(X,d),W_2)$ is a Hadamard ($CAT(0)$) space; but there must be other cases...