Beck's conjecture for annihilating ideal-graphs of commutative rings

Let $R$ be a commutative ring with unity. Define the annihilating ideal-graph $\mathbb{AG}(R)$ to have as vertices the nonzero ideals $I$ with nonzero annihilator, meaning that there is a nonzero ideal $J$ such that $IJ=(0)$; distinct vertices $I$ and $J$ are adjacent precisely when $IJ=(0)$. Write $\chi(\mathbb{AG}(R))$ for its chromatic number and $\operatorname{Clique}(\mathbb{AG}(R))$ for its clique number.

Beck's conjecture. For every commutative ring $R$ with unity,

$$\chi(\mathbb{AG}(R))=\operatorname{Clique}(\mathbb{AG}(R)).$$

The paper states this conjecture as the ring-theoretic claim motivating its extension to multiplicative lattices. The abstract reports a counterexample for multiplicative lattices, while also stating that the conjecture holds for reduced multiplicative lattices; the status of the original ring formulation is not resolved in the supplied text.