Density Bound for Constant Speed Geodesic in 2-Wasserstein Space

Question

Let $\mu_0,\mu_1,\nu$ be continuous probability distributions with full support on $\mathbb{R}^d$ such that $\mathrm{d}\mu_0/\mathrm{d}\nu$ and $\mathrm{d}\mu_1/\mathrm{d}\nu$ are both bounded above by some absolute constant $C$. Let $\mu_t$ denote the unique constant speed geodesic in 2-Wasserstein space from $\mu_0$ to $\mu_1$. Can we say that $\mathrm{d}\mu_t/\mathrm{d}\nu \leq C$ for all $t \in [0,1]$? This statement is true when $\nu$ is the Lebesgue measure instead of a probability measure, according to Proposition 7.29 of Santambrogio's Optimal Transport for Applied Mathematicians, but the proof does not immediately generalize to this setting.