e-positivity of Abreu–Nigro’s g-functions for general q

Prove that for every natural unit interval order m on [n] and every integer 0≤k<n, Abreu–Nigro’s symmetric function g_{m,k}(x;q) is e-positive.

Background

Abreu and Nigro introduced symmetric functions g_{m,k}(x;q) that refine the chromatic quasisymmetric function through a decomposition indexed by k. They conjectured e-positivity for each of these refined functions. The paper proves the conjecture only after specializing q=1, leaving the general q-positivity assertion unresolved.