Closure property of contraction-depth of matroids

The contraction$^*$-depth is the matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph $G$ is the minimum height of a rooted forest whose closure contains $G$ by proving the following for every matroid $M$ (except the trivial case when $M$ consists of loops and bridges only): the contraction$^*$-depth of $M$ plus one is equal to the minimum contraction-depth of a matroid containing $M$ as a restriction.