Chewi–Stromme conjecture on low-temperature log-Sobolev asymptotics

Let $V$ be a potential satisfying a Polyak–Łojasiewicz inequality, and let $f4a4_t$ converge to a probability measure $f4a4_0$ as $t\to0$. Here $C_{\mathrm{LS}}(\mu_t)$ denotes the log-Sobolev constant of $\mu_t$, and $C_{PL}$ is the best constant in the Polyak–Łojasiewicz inequality.

Chewi–Stromme conjecture.

$$\frac{C_{\mathrm{LS}}(\mu_t)-C_{\mathrm{LS}}(\mu_0)}{t}\xrightarrow[]{t\to0^+}2C_{PL}.$$

This conjecture extends the known unique-minimizer result, where the limiting measure is a Dirac mass and the corresponding low-temperature asymptotic is established. It predicts the first-order behavior of the log-Sobolev constant when the zero-temperature limit is a general probability measure.