Bender et al.'s isomorphism conjecture for uniquely complemented zero-divisor graphs

Let $S$ be a commutative semigroup with $0$, and let $G(S)$ be its zero-divisor graph. A graph is uniquely complemented if it is complemented and any two complements of the same vertex have equal neighborhoods. A vertex has a unique complement when it has exactly one complement. Let $\mathcal{P}(n)$ be the semigroup given by the power set of an $n$-element set under intersection.

Bender et al.'s conjecture. If $G(S)$ is uniquely complemented with clique number $3$ or greater and every vertex has a unique complement, then $G(S)$ is isomorphic to $G(\mathcal{P}(n))$, where $n$ is the clique number of $G(S)$.

This is one of three conjectures attributed in the source to C. Bender et al. The supplied material gives no evidence that the statement has been resolved.