The pointed Hopf algebra Dixmier–Moeglin equivalence conjecture

Let $H$ be an affine noetherian pointed Hopf $\mathbb{C}$-algebra, and let $G(H)$ denote its group of grouplike elements. The pointed Dixmier–Moeglin equivalence conjecture. The following are equivalent: (1) $\operatorname{GKdim}(H)$ is finite; (2) $H$ satisfies the Dixmier–Moeglin equivalence; and (3) $G(H)$ is nilpotent-by-finite. The conjecture strengthens the proposed finite-Gelfand–Kirillov-dimension criterion for the Dixmier–Moeglin equivalence from the affine noetherian Hopf setting to the pointed setting. The paper presents it as a conjecture motivated by known results and examples, with no resolution given here.