Birigidity maximality conjecture

Prove that, for all n1,n2,d1,d2, the generic (d1,d2)-birigidity matroid R_{d1,d2}(K_{n1,n2}) is the unique maximal matroid under the weak order among all abstract (d1,d2)-birigidity matroids on E(K_{n1,n2}).

Background

Abstract birigidity extends the gluing axioms to bipartite graphs. The conjecture mirrors Graver’s maximality conjecture but in the birigidity setting. It is known when d1=1 or d2=1 (scene analysis) and when one side is within 3 of full rank via dual uniform matroid products; the general case is open.

A solution would parallel rigidity’s structure theory for bipartite frameworks and inform low-rank matrix completion.