Jackson's conjecture on chromatic roots of 3-connected graphs

Let $G$ be a 3-connected graph that is not bipartite of odd order. A chromatic root is a real or complex zero of the chromatic polynomial $P(G,x)$. Jackson's conjecture. The graph $G$ has no chromatic roots in $(1,2)$. This conjecture concerns extending the chromatic-root-free interval beyond the general bound $32/27$ for sufficiently connected non-bipartite graphs; the source gives no resolution.