Thomassen's conjecture for chromatic roots of 3-connected graphs

Let $G$ be a loopless 3-connected graph with $n$ vertices, and let $P(G,t)$ be its chromatic polynomial. Let $\alpha\approx 1.781$ be the chromatic root of $K_{3,4}$ in $(1,2)$. Thomassen's conjecture.

$$P(G,t)\text{ is non-zero with sign }(-1)^n\quad\text{for }t\in(1,\alpha).$$

If $G$ is not a bipartite graph with an odd number of vertices, the same conclusion should hold throughout $(1,2)$. The second assertion is described as a slight strengthening of a conjecture previously given in the cited source; the proof method is obstructed by edge deletion and contraction producing graphs of connectivity two.