Polynomial- or near-linear-time posterior sampling by annealed Glauber dynamics

Consider the spiked Wigner inference problem with signal-to-noise parameter $\beta_{\bM}$ and threshold $\beta_{\bM}$ as used in the paper. Let $\beta_{\bM}$ denote the scaled posterior parameter, and let $\beta_{\lambda}$ be the threshold appearing in the paper's high-temperature analysis. Annealed Glauber sampling conjecture. For $\beta \le \beta_{\lambda}$, annealed Glauber dynamics samples from the scaled posterior $\mu_{\beta \bM}$ in $\poly(N)$, or even $O(N\log N)$ time. Establishing this would provide a sampling guarantee complementary to the conjectured near-linear-time weak-recovery result; the source describes weak Poincare inequalities and one-dimensional RGD as possible routes, and the conjecture remains open.