The embeddability conjecture for diagram groups and right-angled Artin groups

Let $D(\mathcal{P},w)$ be a diagram group, and let $A(\mathcal{P},w)$ denote the associated right-angled Artin group. A diagram group is embeddable into a right-angled Artin group if it admits an injective homomorphism into some right-angled Artin group. Embeddability conjecture. A diagram group is embeddable into a right-angled Artin group if and only if it does not contain $\mathbb{Z} \wr \mathbb{Z}$. The conjecture proposes an algebraic characterization of those diagram groups that embed into right-angled Artin groups; the preceding discussion notes that the injectivity of the canonical morphism may depend only on the isomorphism class of the diagram group, but no resolution is given here.