Positivity of the chromatic polynomial above maxmaxflow

Let $G$ be a loopless graph with maxmaxflow $\Lambda$, and let $P_G(q)$ be its chromatic polynomial. Maxmaxflow chromatic positivity conjecture. For every real $q>\Lambda$,

$$P_G(q)>0.$$

In particular, $P_G(q)$ has no roots in $(\Lambda,\infty)$. The source presents this as a consequence of the derivative-positivity conjecture; the supplied status evidence reports counterexamples to the stronger preceding conjectures, so this claim is marked refuted.