Let $(X,d)$ be a finite dimensional Alexandrov space of curvature bounded from below. Consider the Wasserstein-2 space over $X$, $(\mathcal{P}_2(X), W_2)$. We know that the subset of compactly supported measures $(\mathcal{P}^c_2(X), W_2)$ is geodesically convex. In a result due S-I-Ohta, in this subset, the angles in the sense of Alexandrov are defined in the strong sense (see: https://www.math.kyoto-u.ac.jp/preprint/2006/19ohta.pdf, theorem 3.5 and remark 3.6). So it is a length space with angles defined in the strong sense. Results in metric geometry show that one can define a notion of differential (directional derivatives) in this subset.
My question is the following: take $X =\mathbb{R}^n$, we know for some $\nu \in \mathcal{P}^c_2(\mathbb{R}^n)$ fixed, the function $\mu \in \mathcal{P}^c_2(\mathbb{R}^n) \mapsto W_2^2(\mu, \nu)$ is locally semiconcave, hence admits a differential. Then is the function $$\mu \mapsto W_2(\mu, \nu)$$ differentiable or semiconcave?