Algebraic and measurable embeddings between square-free right-angled Artin groups

Let $Γ,Δ$ be finite simplicial graphs, and assume that $Γ$ has no induced squares. We prove that the right-angled Artin group $A(Δ)$ measurably embeds into $A(Γ)$ if and only if $A(Δ)$ embeds as a subgroup in a graph product of free abelian groups over $Γ$. This in turn has a graph-theoretical chara