Characterization conjecture for right-angled Artin groups with MCF word problem

Let $\Gamma$ be a finite graph, and let $A(\Gamma)$ be the associated right-angled Artin group. Say that a group has MCF word problem when its word problem is a multiple context-free language. The RAAG MCF-word-problem conjecture. The group $A(\Gamma)$ has MCF word problem if and only if $\Gamma$ is a disjoint union of cliques. The paper says this characterization would follow from the preceding conjecture, together with work of Kropholler, and does not state that it has been proved in the paper.