Lifting annihilators from deletion to the full matroid

Determine whether, for a matroid M on ground set [n], every differential operator ξ ∈ ℝ[∂_1,…,∂_{n−1}] that annihilates the deletion polynomial f_{M \ n} also annihilates the full polynomial f_M, i.e., whether ξ(f_{M \ n}) = 0 implies ξ(f_M) = 0.

Background

This question probes the behavior of annihilators of the basis generating polynomial under deletion, aiming to establish a well-defined map from A(M\n) to A(M) respecting annihilation. A positive answer would underpin the existence of natural morphisms between these Gorenstein rings.