Nonnegative-coefficients conjecture for chain-type polynomials

Let $\tau$ be a chain type, and let $P^\tau\in\mathbb{Z}[z,w]$ be the polynomial associated with $\tau$ in the paper's recursive construction. Nonnegative-coefficients conjecture. For all chain types $\tau$, the polynomial $P^\tau$ has nonnegative coefficients. This conjecture is motivated by computer observations for all chain types of length at most $9$; the general assertion remains unverified in the source.