Non-embeddability conjecture for right-angled Artin groups of tripod trees

Let $T_{p,q,r}$ be the tree with three arms of lengths $p$, $q$, and $r$, and let $P_n$ be the path graph on $n$ vertices. Write $G(_{p,q,r}])$ and $G(P_n)$ for the associated right-angled Artin groups.

Non-embeddability conjecture. If $p,q,r$ are large enough, then $G(T_{p,q,r})$ does not embed into $G(P_n)$ for any $n$.

This proposes a limitation on the family of path graphs as a universal family for right-angled Artin groups. The source does not specify a threshold for “large enough” or provide evidence of a resolution, so the conjecture remains open.