Positivity conjecture for the sixth cumulant of multiple Wiener integrals

Let $X$ be a multiple Wiener--Itô integral of order $p\geq 2$, and let $\kappa_6(X)$ denote its sixth cumulant. Sixth-cumulant positivity conjecture. For every such $X$, one has

$$\kappa_6(X)>0.$$

The conjecture concerns positivity of the sixth cumulant beyond the cases established in the paper, including the two first Wiener chaoses and odd-order integrals under an additional fourth-cumulant condition. The authors explain that their moment-matrix method does not establish positivity in general, so the conjecture remains open.