Universality of the polymatroid quasi-symmetric invariant

Let $\operatorname{PolyMat}$ be the vector space of polymatroids, let $\operatorname{QSym}$ be the quasi-symmetric-function space, and let ${\mathcal G}:\operatorname{PolyMat}\to\operatorname{QSym}$ be the valuative invariant in the preceding theorem. A map $f:\operatorname{PolyMat}\to V$ is valuative when it preserves every rational linear relation among polymatroid base polytopes. Universality conjecture. For every $\mathbb{Q}$-linear valuative map $f:\operatorname{PolyMat}\to V$, there should exist a $\mathbb{Q}$-linear map $\psi:\operatorname{QSym}\to V$ such that

$$\psi\circ {\mathcal G}=f.$$

This asks whether every valuative invariant factors through ${\mathcal G}$; the supplied text does not establish whether this universal property holds.