Conjecture on density of bivariate Tutte-polynomial zeros in uncovered regions

Let $Z_G(q,v)$ be the bivariate Tutte polynomial of a graph $G$, and let regions (a)--(e) be the five real regions defined above.

Zero-density conjecture. As $G$ ranges over all graphs, the zeros of $Z_G(q,v)$ are dense in regions (a)--(e). More precisely, for each fixed $q$ the zeros are dense in $v$, or for each fixed $v$ the zeros are dense in $q$.

If true, this would make the zero-free results in the paper, together with the preceding interval conjectures, best possible in a stronger sense than mere sign variation.