Stembridge's monomial immanant nonnegativity conjecture

Let $A$) be a totally nonnegative $n\times n$ matrix, meaning that all its minors are nonnegative, and let $\lambda\vdash n$ be a partition. For a class function $\phi^{\lambda}$ on $\mathfrak{S}_n$, write $\operatorname{Imm}_{\phi^{\lambda}}(A)$ for its immanant; here the image of $\phi^{\lambda}$ under the characteristic map is the monomial symmetric function $m_{\lambda}$. Stembridge's conjecture.

$$\operatorname{Imm}_{\phi^{\lambda}}(A)\geq 0.$$

This strengthens Schur's and Stembridge's inequalities for immanants and has remained a longstanding open problem in the stated generality.