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Yeah... Perhaps the best bound is in Bolley and Villani (2012) Particular Case 5: $W_p(\mu,\nu)\leq 2^{\frac{1}{2p}} \mathrm{diam}(X)H(\mu||\nu)^{\frac{1}{2p}}$.
The McCann interpolation satisfies the inequalities as equalities for $p$-Wasserstein distance with $p>1$. See Section 7.2 of the book Gradient flows by Ambrosio, Gigli and Savare.
