Conjectural hardness of certification below the spectral threshold for the SK Hamiltonian

Let $\mathsf{M}(\bm W)$ denote the Sherrington–Kirkpatrick Hamiltonian instance with random matrix $\bm W$, and let $c(\bm W)$ be the approximation ratio certified by an algorithm. A certification algorithm runs in polynomial time if its running time is polynomial in the input size. Conjectural hardness of certification. For any $\epsilon > 0$, there does not exist a polynomial-time certification algorithm for $\mathsf{M}(\bm W)$ such that $c(\bm W) \leq 2 - \epsilon$ with high probability. This conjecture predicts that no efficient certification method can asymptotically improve on the spectral threshold for the Sherrington–Kirkpatrick model. The paper presents evidence for this prediction via the degree-4 sum-of-squares relaxation and a low-degree likelihood-ratio argument, conditional on a broader conjecture concerning the validity of low-degree likelihood-ratio analyses for a large class of hypothesis-testing problems.