Conjecture on sublinear chromatic-root bounds for series-parallel graphs

Let $G$ be a finite undirected graph, let $\Delta(G)$ denote its maximum degree, and let $\pi(G,z)$ be its chromatic polynomial.

Series-parallel chromatic-root bound. There exists a universal constant $C<\infty$ such that, for every series-parallel graph $G$ of maximum degree $k$, every chromatic root $z$ lies in

$$|z-1|\leq C\,\frac{k}{\log k}.$$

The paper proves a bound of this order for generalized theta graphs and conjectures that the methods extend to arbitrary series-parallel graphs; it also remarks that an analogous statement for all planar graphs is only conceivable, not asserted.