Nonnegative derivatives of the chromatic polynomial at maxmaxflow

Let $G$ be a loopless graph with maxmaxflow $\Lambda$, and let $P_G(q)$ be its chromatic polynomial. Maxmaxflow derivative-positivity conjecture. The polynomial $P_G(q)/q$ and all its derivatives are nonnegative at $q=\Lambda$; the same then holds for $P_G(q)$ and all its derivatives. The source derives this as a consequence of the preceding half-plane conjecture and reports that the latter has counterexamples, so this conjecture is refuted as part of the same counterexample family.