Behboodi–Rakeei conjecture on the girth of annihilating ideal-graphs

Let $R$ be a reduced ring with more than two minimal primes. The annihilating ideal-graph $\mathbb{AG}(R)$ has girth

$$g r(\mathbb{AG}(R))=3.$$

Behboodi–Rakeei conjecture. Every reduced ring with more than two minimal primes has annihilating ideal-graph of girth $3$. The paper states that this conjecture is settled by proving the corresponding result for reduced multiplicative lattices, so the conjecture is solved.