Conjecture on zero-free intervals for multivariate Tutte polynomials of 2-connected graphs

Let $Z_G(q,\mathbf{v})$ denote the multivariate Tutte polynomial of a graph $G$, let $I_\Diamond(q)$ be the limiting interval defined in the paper, and call an interval self-dual if it has the form $(q/v_m^+(q),v_m^+(q))$. A graph is 2-connected if it remains connected after deletion of any one vertex.

Zero-free interval conjecture. Suppose $0<q<1$. Then there exists a strictly increasing sequence of self-dual intervals ${\mathcal{V}}_m(q)$, $m\geq 2$, such that

$$\lim\limits_{m\to\infty}{\mathcal{V}}_m(q)=I_\Diamond(q),$$

\nand $(-1)^{n-1}Z_G(q,\mathbf{v})>0$ for every 2-connected graph $G=(V,E)$ with $n$ vertices and at least $m$ edges, whenever $v_e\in{\mathcal{V}}_m(q)$ for all $e\in E$.

This would extend the paper's zero-free regions to 2-connected graphs with arbitrarily many edges, with the intervals increasing to the limiting region $I_\Diamond(q)$.