Elgin–Reading–Stella's conjectures on cluster scattering wall-function coefficients

Let $b$ and $c$ be positive integers, and let $\tau(i,j)$ denote the coefficient of $x_1^{ib}x_2^{jc}$ in the wall-function on the ray $\mathbb{R}_{\leq 0}(db,ec)$ of the cluster scattering diagram $\mathfrak{D}_{(b,c)}$, where $d$ and $e$ are coprime positive integers and $(i,j)=(kd,ke)$ for some $k\geq 1$. For the further coefficients $\tau(i,j;k)$ and $\tau^{b,b}(i,j)$, use the notation of the source. Let

$$g=\frac{\gcd(ib,jc)}{\gcd(i,j)}.$$

Elgin–Reading–Stella's conjectures. For $i,j>0$, the coefficient $\tau(i,j)$ satisfies all of the following: it is a polynomial in $b$, $c$, and $g$; this polynomial has $g$ as a factor and degree $\gcd(i,j)$ in $g$, degree $j-1$ in $b$, and degree $i-1$ in $c$; $(\max(i,j))!\tau(i,j)$ has integer coefficients; the formulas C5–C10 in the source hold; $\tau^{b,b}(i,j)$ is a polynomial in $b$ of degree $i+j-1$ with a positive binomial-basis expansion and unimodal log-concave coefficients; the formula C13 holds; and the factorization and polynomiality assertions C14–C18 hold exactly as stated in the source. These include, when applicable, the formulas

$$\tau(1,j)=\frac{g}{b}\binom{b}{j},\qquad \tau(i,1)=\frac{g}{c}\binom{c}{i},$$

and the diagonal formulas C7 and C8. These conjectures concern the dependence of cluster-scattering wall-function coefficients on $b$ and $c$. The paper confirms many of the predictions using quiver representations and tight gradings, while the full collection is not established.