Decidability of the conjugacy problem for general Artin groups

Establish whether the conjugacy problem is decidable for arbitrary Artin groups A(Γ); specifically, determine whether there exists an algorithm that, given two words over the standard generators of A(Γ), decides whether they represent conjugate elements for every Artin group.

Background

Beyond the word problem, the conjugacy problem is a core algorithmic challenge. Although it is solvable for finite type Artin groups (including all braid groups), the general case remains unresolved. The paper emphasizes that this fundamental question is still open and falls outside the scope of the classification results provided for various membership problems.