The Graded Stanley-Stembridge Conjecture

Let $H$ be a type A Hessenberg space. Let $\mathfrak{ch}(\mathrm{L}_{H})$ be the graded character of the associated left representation, and let $\operatorname{Frob}$ denote the Frobenius characteristic map to the symmetric-function ring $\Lambda(x)$. Graded Stanley-Stembridge conjecture.

$$\operatorname{Frob}(\mathfrak{ch}(\mathrm{L}_{H}))$$

is $h_\lambda(x)$-positive. This is the type A positivity statement connected with the Stanley–Stembridge conjecture and with chromatic symmetric functions; the source raises the question of an appropriate type B/C analogue, while no resolution of the displayed statement is given here.