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Polynomially checkable coNP certificate for non-independence in B_{m,n}(2,2)

Identify a coNP certificate for certifying that a bipartite graph is not independent in \mathcal{B}_{m,n}(2,2) that can be verified in polynomial time.

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Background

Even a non-independence certificate verifiable in polynomial time is not known for \mathcal{B}_{m,n}(2,2). The authors point to a candidate proposal in prior work (Jackson–Tanigawa, Conjecture 6.4), but its status remains conjectural.

Establishing such certificates would aid both complexity classification and practical verification in applications involving maximally recoverable tensor codes and related rigidity matroids.

References

We do not even know a coNP certificate, i.e., a certificate that a graph is not independent in $\mathcal{B}{m,n}(2,2)$, that can be checked in polynomial time. A candidate coNP certificate for independence in $\mathcal{B}{m,n}(a,b)$ is given in *{Conjecture 6.4}.

— Rigidity matroids and linear algebraic matroids with applications to matrix completion and tensor codes (2405.00778 - Brakensiek et al., 1 May 2024) in Introduction (following Proposition 1.7)