The random-regular-graph MaxCut pseudoexpectation conjecture

Let $d\geq 3$, and let $G$ be a random $d$-regular graph on $n$ vertices. A pseudoexpectation operator $\widetilde{\mathop{\varmathbb{E}}}$ of degree $n^\delta$ is the degree-bounded SoS relaxation used in the source, acting on Boolean variables $x_i$.

Random-regular-graph MaxCut conjecture. For some $\delta>0$, with high probability there is a degree-$n^\delta$ pseudoexpectation operator $\widetilde{\mathop{\varmathbb{E}}}$ on Boolean variables $x_i$ whose MaxCut value is at least

$$\frac{1}{2}+\frac{\sqrt{d-1}}{d}\bigl(1-\operatorname{o}_{d,n}(1)\bigr).$$

This conjecture concerns the MaxCut setting arising from the proposed planted-problem extension to bottom eigenspaces of random adjacency matrices. It asserts that a low-degree SoS relaxation can certify a value at least the displayed spectral-threshold expression; the source supplies no resolution.