Classify Artin groups with decidable subgroup membership problem

Classify which Artin groups A(Γ) have a decidable subgroup membership problem; that is, determine exactly for which A(Γ) there exists an algorithm that decides, given a finitely generated subgroup H ≤ A(Γ) and an element g, whether g ∈ H.

Background

The paper achieves complete classifications for submonoid and rational subset membership problems in Artin groups via graph-theoretic criteria. In contrast, a corresponding classification for the subgroup membership problem remains unknown. The authors highlight this gap and relate it to specific instances such as the unresolved status for B4 and for the RAAG A(C5).