Divisor class group conjecture for Lovász–Saks–Schrijver rings of forests

Let $G=(V,E)$ be a forest, let $\d=d=\Delta(G)+1$, and let $R_G(d)$ denote its Lovász–Saks–Schrijver ring. Write $\delta(v)$ for the degree of a vertex $v$ in $G$, and set

$$m=\left|\{v\in V:\delta(v)=\Delta(G)\}\right|.$$

Divisor class group conjecture. The divisor class group of $R_G(d)$ is

$$\operatorname{Cl}(R_G(d))\cong\mathbb{Z}^m.$$

This conjecture refines the paper's results on factoriality and divisor class groups for Lovász–Saks–Schrijver rings of forests, including explicit computations for star and path graphs. It predicts that the class group at the normality threshold $d=\Delta(G)+1$ is free abelian with one generator for each vertex of maximum degree; the source provides computational evidence but does not report a proof.