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Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)

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Wasserstein space with strictly non-positive sectional curvature

Let $(X,d)$ be a (separable, complete) metric space with uniformly strictly non-positive curvature in the sense of Alexandrov, i.e. $(X,d)$ satisfies a $CAT(K)$ inequality for some $K<0$. … Does it hold that the 2-Wasserstein space on $(X,d)$ has strictly non-positive curvature? I suspect this is false, but haven't found this result anywhere. …
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