Martsinkovsky's quasi-homogeneity conjecture for the fundamental module

Let $A$ be a normal analytic algebra of Krull dimension two over an algebraically closed field of characteristic zero. The quasi-homogeneity conjecture. The isomorphism

$$D \cong (\Omega_A^1)^{**}$$

is equivalent to $A$ being quasi-homogeneous. This extends the corresponding result for isolated analytic hypersurface singularities and proposes the equivalence for all normal analytic surface algebras in the stated setting.