Isomorphism Between Dual and Standard Artin Groups

Determine whether, for every Coxeter group W and Coxeter element w, the dual Artin group G^*_{W, w} defined from the noncrossing partition poset NC(W, w) via the dual presentation on reflections R_0 is isomorphic to the standard Artin group G_W associated to W.

Background

The dual presentation constructs G^*_{W, w} using maximal chains in NC(W, w) with generators given by reflections labeling cover relations. Bessis proved the isomorphism G^*_{W, w} \cong G_W in spherical cases, and McCammond and Sulway established it in affine cases; it has also been confirmed for rank-three cases.

A uniform understanding across all Coxeter groups would clarify the relationship between combinatorial dual structures and classical Artin presentations, with implications for group theory and topology.