Bender et al.'s reduced-graph conjecture for complemented zero-divisor graphs

Let $S$ be a finite commutative semigroup with zero, and let $G(S)$ be its zero-divisor graph. A graph is complemented if every vertex is orthogonal to some vertex, where adjacent vertices are orthogonal when no third vertex is adjacent to both. Let $\boxempty$ denote the clique number, let $G_r$ be the reduced graph obtained by identifying vertices with the same neighborhood, and let $\mathcal{P}(n)$ be the semigroup of subsets of an $n$-element set under intersection. Bender et al.'s reduced-graph conjecture. If $G(S)$ is a complemented zero-divisor graph with the clique number $n\theseq 3$, then $G(S)$ has reduced graph $G_r$ isomorphic to $G(\mathcal{P}(n))$. The paper gives counterexamples to the motivating conjectures, so this claim is not supported as an open conjecture in the source context.