Generalized Bernstein conjecture for \mathcal{B}_{m,n}(d,d)

Prove that any bipartite graph G that is independent in the bipartite rigidity matroid \mathcal{B}_{m,n}(d,d) admits a d-coloring of its edges that is d-Bernstein (i.e., satisfies the no-monochromatic-cycles condition and admits a vertex labeling satisfying Definition 5.1).

Background

Motivated by Bernstein’s characterization for the (2,2) case, the authors propose a generalization via d-Bernstein colorings. They also show that graphs admitting a d-Bernstein coloring are independent in the dual of \mathrm{T}_{m,n}(m−d,n−d,p) for all characteristics p, so a proof of the conjecture would imply characteristic-independence of these tensor matroids.

This conjecture would provide a unified combinatorial description of independence across all parameters, extending known results and connecting to tropical and matroidal structures.