Weak-order comparability of R_d(K_n) and C^{d−2}_{d−1}(K_n) for d ≥ 4

Determine, for d ≥ 4 and all n, whether the generic d-dimensional rigidity matroid R_d(K_n) is below, above, or incomparable to the generic C^{d−2}_{d−1}-cofactor matroid C^{d−2}_{d−1}(K_n) in the weak order on matroids over E(K_n).

Background

For d ≥ 4, R_d and the cofactor matroid differ (e.g., K_{d+2,d+2} is a circuit in R_d but independent in the cofactor matroid). However, their precise weak-order relation is unknown.

Clarifying this would refine the landscape of abstract rigidity matroids and inform maximality questions.