Goulden–Jackson character nonnegativity conjecture

Let $n$ be a positive integer, let $\mathfrak{S}_n$ be the symmetric group, and let $\chi^{\lambda}$ be any irreducible character of $\mathfrak{S}_n$. For each subinterval $J$ of $[n]=\{1,2,\ldots,n\}$, let $\mathfrak{S}_J$ be the subgroup consisting of permutations that fix every element of $[n]\setminus J$, and define

$$S_J=\sum_{\pi\in\mathfrak{S}_J}\pi$$

as an element of the group algebra of $\mathfrak{S}_n$. Let $\Theta$ be the set of all finite products of elements $S_J$.

Goulden–Jackson conjecture. For every $\theta\in\Theta$, the value of the linear extension of $\chi^{\lambda}$ to the group algebra satisfies

$$\chi^{\lambda}(\theta)\geq 0.$$

This conjecture arose from the study of immanants of Jacobi–Trudi matrices: its truth would imply that those immanants are nonnegative linear combinations of monomial symmetric functions. The supplied text gives no resolution status, so the conjecture is recorded as open.