Maxmaxflow conjecture for bounded chromatic roots

For a graph $G$, define the edge-connectivity between distinct vertices $x$ and $y$ by

$$\lambda(x,y)=\text{the maximum number of edge-disjoint paths from $x$ to $y$}=\text{the minimum number of edges separating $x$ from $y$},$$

and define the maxmaxflow by

$$\Lambda(G)=\max_{x\neq y}\lambda(x,y).$$

Maxmaxflow conjecture. There exist universal constants $C(k)<\infty$ such that every chromatic root $z$ of any graph $G$ with $\Lambda(G)=k$ lies in the disc

$$|z-1|\leq C(k).$$

This weakens a bound in terms of the second-largest vertex degree. The paper notes that linear growth of $C(k)$ in $k$ is natural to expect, but does not prove the conjecture for arbitrary graphs.