Characteristic dependence of matrix completion matroids

Determine whether the symmetric matrix completion matroid \mathcal{S}_n(d), the skew-symmetric matrix completion matroid \mathcal{H}_n(d) (via skew-symmetric matrices of rank ≤ 2d), or the rectangular matrix completion matroid \mathcal{B}_{m,n}(d,d) can depend on the characteristic of the base field, by either constructing a characteristic-dependent example or proving that these matroids are characteristic-independent.

Background

The paper discusses the relationship between linear-algebraic matroids in positive characteristic and matrix completion matroids, including separability issues in characteristic p. While certain tensor and symmetric power matroids do depend on characteristic, the authors note no known examples of characteristic dependence for the matrix completion matroids themselves.

Clarifying this would solidify the robustness of matrix completion matroids across characteristics and refine connections to rigidity and tropical geometry.