A 13-vertex counterexample to $e$-log-concavity for chromatic quasisymmetric functions

We exhibit a connected natural unit interval graph on $13$ vertices whose chromatic quasisymmetric function has an elementary-basis coefficient that is positive, palindromic, and unimodal, but not log-concave.

For the Hessenberg function $h=(2,4,4,6,7,10,10,10,10,12,12,13,13)$ and $\lambda=(6,5,1,1)$, the coefficients of $q^5,q^6,q^7$ in $[e_{\lambda}]X_{G(h)}(\mathbf{x};q)$ are $1,6,38$; hence $6^2<1\cdot38$.

This disproves Conjecture 5.3 of Abreu--Nigro and its later formulations by Tom and by Sagan--Tom.

The calculation is certified by a standalone deterministic verifier using exact integer and rational arithmetic.