Principal representation conjecture for normal homogeneous subrings of graphs

Let $G$ be an unmixed graph, let $S$ be its homogeneous subring, and let $B$ be the configuration defining $S$. Write $(\mathbb{R}_+B)^\circ$ for the relative interior of the cone generated by $B$, and call a representation of $w\in\mathbb{N}B$ principal when it has the principal form defined for the configuration $B$. A strong $\lceil\frac{n}{2}\rceil$-$\tau$-reduction is a reduction of $G$ with the stated strength. Principal representation conjecture. If $S$ is normal and $G$ is unmixed with a strong $\lceil\frac{n}{2}\rceil$-$\tau$-reduction, then every

$$w\in(\mathbb{R}_+B)^\circ\cap\mathbb{N}B$$

has a principal representation. This representation property is intended to support the characterization of when the normal homogeneous subring $S$ is Gorenstein; the paper subsequently uses it to establish the corresponding implication, while the conjectural equivalence itself is the broader claim left for verification.