Gorenstein characterization conjecture for normal homogeneous subrings of graphs

Let $G$ be a graph with homogeneous subring $S$, and assume that $S$ is normal. Say that $G$ is unmixed when all its relevant minimal vertex covers have the same cardinality, and let a strong $\lceil\frac{n}{2}\rceil$-$\tau$-reduction mean a strong reduction of the indicated size. Gorenstein characterization conjecture. Under the assumption that $S$ is normal,

$$S\text{ is Gorenstein}\quad\Longleftrightarrow\quad G\text{ is unmixed with a strong }\left\lceil\frac{n}{2}\right\rceil\text{-}\tau\text{-reduction}.$$

This connects the commutative-algebraic property of being Gorenstein with a combinatorial reduction of the graph. The supplied parser gives no resolution evidence, so the conjectural characterization remains open here.