Existence of a natural ring homomorphism from A(M\e) to A(M)

Ascertain whether, for every matroid element i ∈ E, there exists a graded ring homomorphism θ: A(M \ i) → A(M), where A(·) denotes the Gorenstein ring associated to the basis generating polynomial, compatible with the matroid deletion operation.

Background

To pursue a semi-small type decomposition of A(M) analogous to those known for Chow and augmented Chow rings of matroids, a structural map from deleted matroid rings A(M\i) into A(M) would be fundamental. The author raises the question of whether such a graded ring homomorphism exists for all deletions.