The parameter-dependence conjecture for peak-count positivity

Let $\mathcal{V}$ be a positive vertical path with $u$ up-steps and $d$ down-steps, let $Q$ be the set of quarter-planar paths with $r$ right-steps, $l$ left-steps, and vertical projection $\mathcal{V}$, and let $G_1$ be the peak-count generating function over $Q$. A polynomial is $(x+1)$-positive when it belongs to $\mathbb{N}_0[x+1]$. The parameter-dependence conjecture. Whether $G_1$ is $(x+1)$-positive depends only on $r$, $l$, $u$, and $d$. The claim is presented as a consequence of the equivalence conjecture together with the theorem that $G_2$ depends only on these four step counts; the paper notes that $G_1$ itself need not depend only on them.