The ordinary-flat conjecture for complex-representable matroids

Let $M$ be a matroid, and call a rank-$k$ flat ordinary if it is the direct sum of a point and a rank-$(k-1)$ flat. Ordinary-flat conjecture. For each integer $k\ge 2$, every complex-representable matroid with rank at least $k+2$ has an ordinary rank-$k$ flat. This conjectures the complex-representable analogue of Hansen's theorem for real-representable matroids. Bounds of order $\Omega(k)$ were previously known, and the paper proves the weaker bound $4(k-1)$; the conjectured bound remains open.