Wasserstein distance of convolution of measures

Question

Let $\mu_1, \mu_2,\nu_1,\nu_2$ be measures on $\mathbb{R}^d$. It is well known that the $p$-Wasserstein distance satisfies $$\mathcal{W}_p(\nu_1 *\mu_1, \nu_1 *\mu_2) = \mathcal{W}_p(\mu_1,\mu_2),$$ by considering the coupling formulation. I am wondering whether this result can be extended and whether one can say $$\mathcal{W}_p(\nu_1 *\mu_1, \nu_2 *\mu_2) \leq \mathcal{W}_p(\nu_1,\nu_2) + \mathcal{W}_p(\mu_1,\mu_2).$$ Taking $\nu_1=\nu_2$ gives us the original expression, so this is a proper generalization. Is this in fact true? I have had some trouble proving it using the coupling formulation, so either a proof or counterexample is of interest to me.