Log-concavity of elementary coefficients for low-rank abelian Hessenberg graphs, with a counterexample in general
Let $X_{G_h}(\mathbf{x};q)=\sum_{\mu\vdash n}c_\mu(q)e_\mu(\mathbf{x})$ be the chromatic quasisymmetric function of the natural unit interval graph attached to a Hessenberg function $h$. We establish an infinite class, valid in all orders, for which every nonzero polynomial $c_\mu(q)$ has a nonnegative, log-concave coefficient sequence with interval support. Namely, this holds whenever $h$ is abelian and its complement-Ferrers partition $\lambda$ satisfies $\min\{\lambda_1,\ell(\lambda)\}\leq 3$; equivalently, the diagram has at most three rows or at most three columns. Cubic interpolation reduces the rank-three case to a uniform theorem for a difference of two products of four $q$-integers, proved by positive decomposition, interval methods, and finite-window smoothing. The argument also yields explicit formulas for every supported elementary coefficient in complement-Ferrers rank at most three. We also include a connected 13-vertex natural unit interval graph for which one elementary coefficient is positive, palindromic, and unimodal but not log-concave, thereby recording the failure of the unrestricted conjecture. Thus low complement-Ferrers rank gives a substantial positive regime even though coefficientwise $e$-log-concavity fails in general.