$(a+1)$-positivity conjecture for quarter-plane loop series

Let $Q(a,u)$ be the generating series for quarter-plane loops. $(a+1)$-positivity conjecture. The series $Q(a,u)$ can be expanded as

$$Q(a,u)=\sum_{n\ge 0}u^nP_n(a+1),$$

where $P_n(x)\in\mathbb{N}[x]$. Thus, the coefficients of $Q(a,u)$, when expressed in powers of $a+1$, are non-negative. This had been checked computationally through half-length $n=100$, but the general claim was unproved in the source.