Sharp Poincaré interpolation along Wasserstein geodesics

Let $μ_0$ and $μ_1$ be $κ_0$- and $κ_1$-strongly log-concave probability measures on $\R^n$, and let $(μ_t)_{t\in[0,1]}$ be their quadratic Wasserstein geodesic. We prove the sharp Poincaré constant estimate \[ \sqrt{C_P(μ_t)} \leq \frac{1-t}{\sqrt{κ_0}} + \frac{t}{\sqrt{κ_1}}. \] The coefficient is