The equivalence conjecture for ordinary and signed 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 $G_1$ and $G_2$ be respectively the peak-count and signed peak-count generating functions over $Q$. A polynomial is $(x+1)$-positive when it belongs to $\mathbb{N}_0[x+1]$. The equivalence conjecture. The polynomial $G_1$ is $(x+1)$-positive if and only if $G_2$ is $(x+1)$-positive. The conjecture was tested for $u,d,r,l\leq 5$ and is proposed because the signed generating function has an explicit parameter dependence; proving it would imply the preceding positivity conjecture.