The constant-degree sum-of-squares barrier conjecture for random matrices

Fix $1<\alpha<2$, set $T=N^\alpha$, and let $M_{ran}$ be a random $T\times N$ Gaussian matrix. Let $v\in\mathbb R^N$ satisfy $\|v\|_\infty\leq1$, and suppose $|(M_{ran}v)_j|\geq N^\sigma$ for every $j\in W$. Constant-degree sum-of-squares barrier conjecture. If $\sigma\leq1-\alpha/4$ and $c>0$, then the degree-$O(1)$ sum-of-squares method cannot prove

$$|W|\lesssim N^{\alpha+1-2\sigma-c}.$$

The claim is attributed to the cited work of D'Hart and collaborators and is presented as a conjectural limitation of constant-degree sum-of-squares proofs; the supplied text gives no resolution.