Coefficient-positivity conjecture for the tau-deformed polynomials

Let $\pi$ be a matching, let $d(\pi)$ be the degree parameter used in the source, and let $P_\pi(\tau,t)$ be the associated bivariate polynomial. Tau-deformed coefficient-positivity conjecture. The bivariate polynomial

$$d(\pi)!P_\pi(\tau,t)$$

has nonnegative integer coefficients. This is the tau-deformed analogue of coefficient positivity; the source presents it as conjectural and gives computational evidence, but no general proof.