Woodall's conjectures on chromatic roots of highly connected plane triangulations

A plane triangulation is a loopless plane graph in which every face has size three. Let $P(G,t)$ be its chromatic polynomial, let $\tau=(1+\sqrt{5})/2$ be the golden ratio, and let $\theta\approx2.61819$ be the chromatic root of the icosahedron in $(2,3)$. Woodall's conjectures.

1. If $G$ is 4-connected, then $P(G,t)$ has at most one zero in $(2,\tau^2)$.
2. If $G$ is 5-connected, then $P(G,t)$ has exactly one zero in $(2,\theta)$ and no zeros in $(\theta,3)$.

These conjectures seek extensions of the known zero-distribution theorem for 3-connected plane triangulations. The source also gives a related finiteness conjecture: for every $\epsilon>0$, only finitely many 4-connected plane triangulations have a chromatic root in $(2,\tau^2-\epsilon)$, and only finitely many 5-connected ones have a root in $(\tau^2+\epsilon,3)$.