Strong matroid covering bound

Prove that for every finite set V of dimension vectors, each having at most t coordinates, and k equal to the maximum coordinate among the vectors in V, the matroid covering number satisfies C_M(V) <= binom(t+k+1,k+1).

Background

The paper establishes the bound C_A(V) <= binom(t+k+1,k+1) for representable configurations over fields, using an algebraic argument involving spaces of low-degree polynomials. For arbitrary matroids, the authors obtain only the weaker estimate C_M(V) <= sum_{i=0}^{k+1} t^i through a combinatorial argument.

The conjecture asks whether the sharper representable bound remains valid for all matroids, including non-representable matroids. A proof would extend the principal quantitative covering result beyond the representable setting and substantially improve the general matroid estimate.