The positive graph conjecture

Let $H$ be a graph. Its homomorphism density in a bounded measurable symmetric kernel $W:[0,1]^2\to\mathbb{R}$ is

$$t_H(W)=\int_{[0,1]^{V(H)}}\prod_{ij\in E(H)}W(x_i,x_j)\,d\mu^{V(H)},$$

where $bc$ is Lebesgue measure. An automorphism $\phi$ of $H$ is a stable involution if it is an involution for which there is a partition $L\cup R\cup F$ of $V(H)$, with $F$ a vertex cut separating $L$ and $R$, such that $\phi$ exchanges $L$ and $R$, fixes $F$, and $F$ is an independent set. A graph is positive if $t_H(W)\geq 0$ for every such kernel $W$.

The positive graph conjecture. A graph $H$ is positive if and only if it has a stable involution.

The forward implication would characterize all graphs whose homomorphism densities are non-negative on arbitrary kernels; the reverse implication follows from the Cauchy--Schwarz inequality. The conjecture is also viewed as an analogue of Hilbert's seventeenth problem, with stable-involution graphs playing the role of squares. No resolution is supplied in the source.