Isomorphism between the graded Möbius algebra and the Gorenstein ring in low degrees

Determine whether, for every matroid M of rank d, the natural map from the graded Möbius algebra H^k(M) to the Gorenstein ring A^k(M) induced by the basis generating polynomial is an isomorphism in degrees k ≤ d/2.

Background

The graded Möbius algebra H(M) and the Gorenstein ring A(M) are closely related but generally distinct. Establishing an isomorphism in low degrees would clarify their relationship and could enable transferring structural decompositions and Hodge-theoretic properties between them.