The excluded minors for $k$-polymatroids with binary $k$-natural matroids

If $\mathcal{C}$ is a minor-closed class of matroids, then the class ${\tilde{\mathcal{C}}}_k^{\mathrm{\prime }}$ of $k$-polymatroids whose $k$-natural matroids are in $\mathcal{C}$ is also minor-closed. We investigate the following question: When $\mathcal{C}$ is the class of binary matroids, what are the excluded minors for ${\tilde{\mathcal{C}}}_k^{\mathrm{\prime }}$? When $k=1$, ${\tilde{\mathcal{C}}}_1^{\mathrm{\prime }}$ is simply the class of binary matroids, which has $U_{2,4}$ as its only excluded minor. Joseph E. Bonin and Kevin Long answered the question for $k=2$ and found that the set of excluded minors for ${\tilde{\mathcal{C}}}_2^{\mathrm{\prime }}$ is infinite. We determine the sets of excluded minors for ${\tilde{\mathcal{C}}}_k^{\mathrm{\prime }}$ when $k\ge 3$ and find that they are finite. There are $12$ excluded minors for ${\tilde{\mathcal{C}}}_3^{\mathrm{\prime }}$ and when $k>3$, there are $k+7$ excluded minors for ${\tilde{\mathcal{C}}}_k^{\mathrm{\prime }}$.