The polynomiality and positivity conjectures for the character-variety coefficients

Let $\overline H_n(z,w)=\sum_{i,j}\overline h^n_{i,j}z^jw^{-i}$ be the Laurent expansion associated with the mixed Hodge generating function.

Polynomiality and positivity conjectures. The following two assertions are conjectured:

1. $\overline H_n(z,w)$ is a polynomial in $z,w$.
2. The coefficients $(-1)^j\overline h^n_{i,j}$ of $\overline H_n(z,-w)$ are non-negative integers.

These are combinatorial consequences of the main mixed-Hodge polynomial conjecture and are not established in the supplied text.