Chromatic roots and minor-closed families of graphs

Given a minor-closed class of graphs $\mathcal{G}$, what is the infimum of the non-trivial roots of the chromatic polynomial of $G\in \mathcal{G}$? When $\mathcal{G}$ is the class of all graphs, the answer is known to be $32\mathrm{/}27$. We answer this question exactly for three minor-closed classes of graphs. Furthermore, we conjecture precisely when the value is larger than $32\mathrm{/}27$.