Two-sided limiting-threshold conjecture for the symmetric binary perceptron

For $\kappa>0$, define the symmetric binary-perceptron $m$-OGP threshold $\alpha_m^*(\kappa)$ as the infimum density at which the appropriate $m$-OGP occurs, and set

$$\alpha_\infty^*(\kappa)=\lim_{mtoinfty}\alpha_m^*(\kappa).$$

Two-sided limiting-threshold conjecture. For every $\epsilon>0$, there exists $\kappa^*(\epsilon)>0$ such that for every $kappalekappa^*(\epsilon)$: no polynomial-time search algorithm for the SBP exists if $\alpha>(1+\epsilon)\alpha_\infty^*(\kappa)$, while a polynomial-time search algorithm exists if $\alpha<(1-\epsilon)\alpha_\infty^*(\kappa)$. This conjecture would identify the limiting $m$-OGP threshold with the algorithmic threshold up to arbitrarily small relative error as $\kappa\to0$.