Conjecture on sign variation for multivariate Tutte polynomials in uncovered regions

Let $Z_G(q,v)$ be the bivariate Tutte polynomial of a graph $G$. Consider the following regions in the real $(q,v)$-plane: (a) $q<0$ and $v<-2$; (b) $q<0$ and $0<v<-q/2$; (c) $0<q\leq32/27$, $q\neq1$, and $v<v_\Diamond^-(q)$; (d) $0<q\leq32/27$, $q\neq1$, and $v_\Diamond^+(q)<v<0$; or (e) $q>32/27$ and $v<0$.

Sign-variation conjecture. Fix real $q$ and $v$ in one of these regions. Then, for all sufficiently large $n$ (depending on $q$ and $v$) and all sufficiently large $m$ (depending on $q$, $v$, and $n$), there exist 2-connected graphs $G$ with $n$ vertices and $m$ edges for which $Z_G(q,v)$ is nonzero and takes either sign.

This conjecture asserts that sign control cannot be extended to the listed regions using only the numbers of vertices and edges. The paper further suspects that, for $0<q\leq32/27$ and perhaps up to $q=2$, the graphs can be chosen to be series-parallel, while known positivity results restrict such a choice when $q>2$ and $v\geq-1$.