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

Let $Q(a,s,u)$ be the generating function for quarter-plane loops, refined by the variable $s$, and let $Q^{\bullet}(a,s,u)$ be the corresponding generating function counting primitive quarter-plane loops. Refined $(a+1)$-positivity conjecture. Both series $Q(a,s,u)$ and $Q^{\bullet}(a,s,u)$ are $(a+1)$-positive; that is, their coefficients, when expressed in powers of $a+1$, are non-negative. This is a refinement of the univariate $(a+1)$-positivity property, and the source gives no proof of the refined claim.