Branch-depth as minor closure of contraction-deletion-depth

A matroid is a finite set equipped with a rank function satisfying the matroid axioms. Its branch-depth and contraction-deletion-depth are the graph- and matroid-depth parameters considered in the source. For a class of matroids, being minor-closed means that it contains every minor of each of its members.

Branch-depth minor-closure conjecture. There exists a function $f$ such that, if a matroid $M$ has branch-depth $d$, then there exists a matroid $N$ with contraction-deletion-depth at most $f(d)$ such that $M$ is a minor of $N$.

The result is proved in the paper for representable matroids, with an explicit bound on the contraction-deletion-depth of $N$. The conjecture asks whether the analogous statement holds for all matroids, possibly with a worse dependence on branch-depth.