Classify Artin groups with decidable group and identity problems

Classify all Artin groups A(Γ) for which the group problem (deciding, on input a finite set X of words, whether Sgp(X) is a subgroup of A(Γ)) and the identity problem (deciding, on input a finite set X of words, whether 1 ∈ Sgp(X)) are decidable.

Background

The paper provides complete classifications for several membership-type decision problems (e.g., submonoid and rational subset membership) in Artin groups via forbidden induced subgraphs of the defining graph. However, analogous classifications for the group problem and the identity problem are not yet known. The authors provide partial negative results for some families and show related fixed-target problems can be undecidable, but a full classification remains to be determined.