Six-cycle link criterion conjecture for relative Artin complexes

Let $\Lambda$ and $\Lambda'$ satisfy the assumptions of the source's six-cycle link lemma, with $\Lambda'$ not necessarily admissible in $\Lambda$. Consider the links of vertices in the relative Artin complex $\Delta_{S,S'}$. Six-cycle criterion conjecture. There exists a criterion involving only cycles of length at most $6$ in the $1$-skeletons of these vertex links such that, whenever the criterion is satisfied, $\Delta_{S,S'}$ is contractible. The conjecture seeks to extend the existing criterion beyond admissible subgraphs and remains open.