Non-realizable extension of the nonnegativity corollary

Let ${\bf X}=(X,\operatorname{rk})$ be a polymatroid, and let ${\mathcal H}[{\bf X}]$ and ${\mathcal P}[{\bf X}]$ denote the invariants appearing in Corollary

$$. **Non-realizable extension conjecture.** Corollary$$

should remain true even when ${\bf X}$ is a polymatroid that is not realizable. The claim proposes that the three stated nonnegativity assertions for the specializations of ${\mathcal H}[{\bf X}]$ and ${\mathcal P}[{\bf X}]$ extend from realizable polymatroids to all polymatroids; no proof or resolution is supplied here.