The projective-plane core conjecture for minimally nonideal matrices

A minimally nonideal matrix is a clutter matrix that is not ideal but whose proper minors are all ideal. Its core is the submatrix consisting of its minimum-weight rows. A non-degenerate projective plane is viewed as a clutter whose vertices are points and whose hyperedges are lines. Projective-plane core conjecture. If $A$ is a minimally nonideal matrix whose core is a non-degenerate projective plane, then $A$ is square and equal to the point-line incidence matrix of the Fano plane. The conjecture asks whether the highly structured non-degenerate projective-plane case admits any minimally nonideal matrices beyond the Fano-plane incidence matrix; the paper presents this as an open classification problem.