e-positivity conjecture for the symmetric functions g_k of Hessenberg graphs

Let $\mathbf{m}\colon[n]\to[n]$ be a Hessenberg function, let $G_\mathbf{m}$ be its indifference graph, and let $g_k(G_\mathbf{m};x,q)$ be the symmetric functions defined in the paper. A symmetric function is e-positive if it is a nonnegative linear combination of elementary symmetric functions. e-positivity conjecture for the functions $g_k$. The symmetric functions $g_k(G_\mathbf{m};x,q)$ are e-positive for every Hessenberg function $\mathbf{m}\colon[n]\to[n]$ and every $k=0,\ldots,n-1$.

This conjecture is motivated by the identity expressing the chromatic quasisymmetric function in terms of the functions $g_k$: e-positivity of all the $g_k$ would imply e-positivity of the chromatic quasisymmetric function and therefore the Stanley–Stembridge conjecture. The supplied text gives no resolution status.