Maximal tensor products of uniform matroids and birigidity duality

Prove that the family of tensor products of two uniform matroids has a unique maximal element under the weak order and that this unique maximal element is the dual of the generic birigidity matroid R_{d1,d2}(K_{n1,n2}).

Background

Brakensiek et al. identified the dual of the generic (d1,d2)-birigidity matroid as a tensor product of uniform matroids. The conjecture asks for uniqueness of a maximal such tensor product, paralleling Mason’s symmetric-power question.

This would yield a dual maximality principle for birigidity akin to that conjectured for rigidity.

References

Conjecture\nThere is a unique maximal element in the family of tensor products of two uniform matroids.\nMoreover, this unique maximal element is the dual of the generic birigidity matroid.

— Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach  (2508.11636 - Cruickshank et al., 29 Jul 2025) in Birigidity

However, the natural case of $M_1$ and $M_2$ being uniform matroids remains open, and its restriction to tensor products has established itself in modern literature under the name Mason's conjecture (see).

— Linear matroid products: a synthetic approach  (2609.16747 - Tyomkyn, 15 Sep 2026) in Section 1, Introduction, immediately before Conjecture (Mason’s conjecture)