  DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a
natural analogue of tree-depth of graphs. They conjectured that a matroid of
sufficiently large branch-depth contains the uniform matroid $U_{n,2n}$ or the
cycle matroid of a large fan graph as a minor. We prove that matroids with
sufficiently large branch-depth either contain the cycle matroid of a large fan
graph as a minor or have large branch-width. As a corollary, we prove their
conjecture for matroids representable over a fixed finite field and
quasi-graphic matroids, where the uniform matroid is not an option.
