Subexponential optimization and sampling conjecture for the low-temperature CREM

Consider a continuous random energy model (CREM) with concave covariance function $A$, inverse temperature $\beta>\beta_c$, \tilde\text{ and } $\text{?}$, and $\text{?}$. Let $\text{OPT}=\text{max}_{|v|=N}X_v$ and let $\text{KL}$ denote Kullback–Leibler divergence. Low-temperature algorithmic tradeoff conjecture. For $\text{?}$ and $\text{?}$, there is a $2^{\widetilde{O}_{\varepsilon}(N^\alpha)}$-time algorithm which, with high probability, both finds $\widehat v$ satisfying $X_{\widehat v}\geq\mathrm{OPT}-\varepsilon N^{1-2\alpha}$ and outputs a distribution $\widehat\mu$ satisfying $\operatorname{KL}(\widehat\mu\|\mu_{\beta,N})\leq\varepsilon N^{1-2\alpha}$. The conjecture proposes a Pareto tradeoff between running time and optimization and sampling accuracy in the low-temperature regime, while the supplied text does not specify whether it has been proved or disproved.