Positivity and sum-rule conjecture for tau-deformed special values

For a matching $\pi$, define $G_\pi(\tau)=\Psi_\pi(\tau,-|\pi|)$. Let $d(\pi)$ be the degree parameter used in the source, and let $\Psi_\pi(-\tau)$ denote the specialization $\Psi_\pi(-\tau,0)$. Tau-deformed positivity and sum-rule conjecture.

$$G_\pi(\tau)=(-1)^{d(\pi)}g_\pi(\tau),$$

where $g_\pi(\tau)$ is a polynomial with nonnegative integer coefficients, and

$$\sum_\pi G_\pi(\tau)=\sum_\pi\Psi_\pi(-\tau).$$

These conjectural properties were checked in the computed range $|\pi|\leq8$; their general validity remains open.