The toggle-buildability conjecture for signed peak-count generating functions

Let $G_2$ be the signed peak-count generating function for the quarter-planar paths under consideration. Define $B=\{1\}\cup\{\operatorname{Bin}^{j}(2j)/2:j\in\mathbb{N}\}=\{1\}\cup\{\operatorname{Bin}^{j}(2j-1):j\in\mathbb{N}\}$, and call a polynomial toggle-buildable if it is a linear combination of elements of $B$ with non-negative coefficients. A polynomial is $(x+1)$-positive when it belongs to $\mathbb{N}_0[x+1]$. The toggle-buildability conjecture. The generating function $G_2$ is toggle-buildable exactly when it is $(x+1)$-positive. The conjecture was tested for $r,l,u,d\leq 5$ and concerns whether two positivity descriptions of the signed peak-count polynomial coincide.