Existence of other dihedral dual reflection groups

Determine whether there exists any integer n, with n ≠ 4, such that the dihedral group D_{2n} (of order 2n) coacts homogeneously and inner faithfully as a dual reflection group on a noetherian Artin–Schelter regular domain A generated in degree 1, with the identity component A_e also Artin–Schelter regular. Equivalently, classify all n for which D_{2n} can act as a dual reflection group beyond the known case D_8.

Background

In the paper, the authors present an explicit example where the dihedral group D_8 (of order 8) coacts as a dual reflection group on a 3-dimensional AS regular algebra, with the identity component being AS regular. This comes with a concrete grading and explicit invariant subring.

The authors note that outside D_8, it is unclear whether any other dihedral group D_{2n} can serve as a dual reflection group under the same framework. The problem asks to extend or refute this to general n by identifying dual reflection structures or proving nonexistence.