Jackson–Sokal's density conjecture for Tutte-polynomial zeros

Let $G$ be a graph with vertex set $V$ and edge set $E$, and let $Z_G(q,v)=\sum_{A\subseteq E}q^{k(A)}v^{|A|}$ be its Tutte polynomial, where $k(A)$ is the number of components of $(V,A)$. Let $v^+_\lozenge(q)$ be the middle branch of the curve $v^3-2qv-q^2=0$ for $0<q<32/27$, and define

$$v^-_\lozenge(q)=\frac{q}{v^+_\lozenge(q)}.$$

Jackson–Sokal's density conjecture. The zeros of the Tutte polynomials of graphs are dense in the following regions: (a) $q<0$ and $v<-2$; (b) $q<0$ and $0<v<-q/2$; (c) $0<q\leq 32/27$ and $v<v^-_\lozenge(q)$; (d) $0<q\leq 32/27$ and $v^+_\lozenge(q)<v<0$; and (e) $q>32/27$ and $v<0$. The statement is presented in the source as Jackson and Sokal's conjecture; the supplied text does not indicate whether it has been resolved.