The linear-or-quadratic distortion conjecture for generalised Bestvina–Brady subgroups

Let $\Gamma$ be a simplicial graph, let $A_\Gamma$ be its associated right-angled Artin group, and let $H_\Phi$ be a finitely generated generalised Bestvina–Brady subgroup of $A_\Gamma$.

Linear-or-quadratic distortion conjecture. The distortion of $H_\Phi$ in $A_\Gamma$ is linear or quadratic.

The claim predicts that every finitely generated generalised Bestvina–Brady subgroup has one of these two distortion types. The supplied passage gives examples and surrounding results but does not establish the assertion in full, so its resolution is not determined there.