Slow-thermalization conjecture at the dynamical transition

Let $P_t$ be the Langevin semigroup, let $dx$ be the uniform measure, and let $\pi_{T,N}$ be the equilibrium measure at temperature $T$. Define the thermalization time of the uniform measure by

$$\tau_* = \inf\left\{t:\max_{\lVert f\rVert_\infty\leq 1}\int\left(P_t f-\int f\,d\pi_{T,N}\right)^2dx\leq\frac{1}{e}\right\}.$$

Slow-thermalization conjecture. Starting from the uniform measure, Langevin dynamics takes exponential time to reach equilibrium for every $T<T_{sh}$: there is a constant $c>0$ such that

$$\tau_*\geq e^{cN}$$

with probability tending to $1$. For $T>T_{sh}$,

$$\tau_*=O(1).$$

The conjecture gives a dynamical interpretation of the shattering temperature as the onset of slow thermalization from a uniformly random start, rather than necessarily as a threshold for slow mixing. The source explicitly cautions that the exact formulation may not be correct as stated.