Front-loading of the differential defect

Prove that for every simple rank-r matroid M, the differential defect \delta_k(M)=h_k^\beta(M)-h_k^0(M) satisfies \delta_k(M)\ge\delta_{r-k}(M) for every k\le r/2, equivalently h_k^\beta(M)\ge h_{r-k}^\beta(M) for every k\le r/2.

Background

The classical apolar Hilbert function h_k^0(M) is symmetric, while the enlarged Hilbert function h_k^\beta(M) incorporates the additional lowering operator D_\beta. The quantity \delta_k(M) measures the extra contribution supplied by this differential completion in degree k.

The conjecture asserts that this additional contribution is front-loaded toward lower differential degrees. Exact computation verifies the inequality for all 950 simple matroids on eight elements and for the reported modular sample of 400 simple nine-element matroids, but the paper provides no general proof or map between complementary defect spaces.