Graver’s maximality conjecture for d = 3

Prove that for every n, the generic 3-dimensional rigidity matroid R_3(K_n) is the unique maximal matroid under the weak order among all abstract 3-rigidity matroids on E(K_n).

Background

The conjecture is true for d=2 (Geiringer/Laman) and false for d ≥ 4 (counterexamples via K_{d+2,d+2}); it is proved in this paper that the C_2^1-cofactor matroid is the unique maximal abstract 3-rigidity matroid, but whether R_3 itself is maximal remains open.

A positive resolution would yield a combinatorial rank characterization for 3D rigidity via cover formulas established for the cofactor matroid.