Stable-algorithm existence conjecture for the symmetric binary perceptron

Let $\kappa>0$ and let the symmetric binary perceptron (SBP) have constraint density $\alpha$. A search algorithm is stable when it satisfies the stability notion used in the paper. Stable-algorithm conjecture. There exists a stable algorithm that finds an SBP solution with high probability when

$$\alpha=O(\kappa^2).$$

This conjecture asks whether the low-density algorithmic regime can be achieved by a stable algorithm, paralleling the established stability result for the Kim–Roche algorithm in the asymmetric model.