Linear chromatic-root bound in maxmaxflow

For a loopless weighted graph $G$, let $P_G(q)$ be its chromatic polynomial, let its chromatic roots be the real or complex zeros of $P_G$, and let $\Lambda$ be its maxmaxflow.

Maxmaxflow chromatic-root conjecture. There exist universal constants $C(\Lambda)<\infty$ such that every chromatic root of every loopless graph of maxmaxflow $\Lambda$ lies in

$$|q|\leq C(\Lambda).$$

Moreover, $C(\Lambda)$ can be taken to be linear in $\Lambda$. The conjecture is the paper's principal motivation for studying maxmaxflow; the source gives no resolution.