The doubly nonnegative function and good graph conjecture

Let $\mathcal{G}_+$ be the set of bounded measurable nonnegative symmetric functions on $[0,1]^2$, and let $\mathcal{H}$ be the space of bounded measurable real functions on $[0,1]^2$. A function $g\in\mathcal{G}_+$ is doubly nonnegative if there is an $h\in\mathcal{H}$ such that

$$g(x,y)=\int_{[0,1]}h(x,z)h(y,z)\\,d\mu(z),$$

equivalently, if it is nonnegative, symmetric, and has nonnegative spectrum. A function $g$ is nice if $t(G,g)\geq\lVert g\rVert^{{\rm e}(G)}$ for every simple graph $G$, and a simple graph $G$ is good if this inequality holds for every doubly nonnegative $g$.

The doubly nonnegative function and good graph conjecture. All doubly nonnegative functions are nice, and all simple graphs are good.

These two assertions concern the inequality relating graph homomorphism integrals to the mean of the function. The paper establishes that all graphs with at most five vertices are good and proves several further families, but the assertions in full remain open.