Set-weighted chromatic symmetric function coefficient-sum conjecture

Let $(G,\omega)$ be a set-weighted graph with $n$ vertices and total weight $d$, and write

$$X_{(G,\omega)}=\sum_{\lambda\vdash d}c_{\lambda}e_{\lambda}.$$

Let $\mu\leq d$ be an integer, viewed as a one-part partition, that is $s$-allowable in $(G,\omega)$, and fix $j\in\{0,\dots,d-\mu\}$, with $j=0$ permitted only when $\mu=d$. For an acyclic orientation $\gamma$ and a generalized $2$-step weight map $S$, let $\operatorname{swts}(\gamma,S)$ and $\operatorname{wts}(\gamma,S)$ denote the associated statistics, and let $\operatorname{sgn}(\gamma,S)$ be the sign defined from the first nonempty step of each relevant stable set. Set-weighted coefficient-sum conjecture.

$$\sigma_{\mu,j}\bigl(X_{(G,\omega)}\bigr)=(-1)^{d-n}\sum_{\substack{\operatorname{swts}(\gamma,S)=(\mu,j)\\ S\ \operatorname{admissible}}}\operatorname{sgn}(\gamma,S),$$

summed over all acyclic orientations $\gamma$ of $G$ and all $\gamma$-admissible generalized $2$-step weight maps $S$ of $G$ such that $\operatorname{wts}(\gamma,S)=(\mu,j)$. The conjecture gives a combinatorial interpretation of a broad family of sums of $e$-basis coefficients of vertex-weighted chromatic symmetric functions. It extends a known follow-up result of Stanley; in particular, the abstract states that it would give an interpretation for sums with prescribed values of $\lambda'_1$ and $\lambda'_2$ for every unweighted claw-free graph. The source supplies no resolution.