Minkowski-relation presentation of the face-part cohomology

Let $X$ be a torus manifold with orbit space $X/T$, let $K$ be the associated simplicial complex, and let $x_I$ denote the face-part cohomology class corresponding to a simplex $I\in K$. For $|I|=k$, let $X_I/T$ be the corresponding orbit-space face, let $\lambda(I)_\mu$ denote the characteristic-map coefficient indexed by $\mu\in\Lambda^kV$, and let $a$ range over functions on the $k$-simplices such that the indicated chain bounds in $C_{n-k}(X/T;\mathbb{R})$.

Minkowski-relation presentation conjecture. As a vector space, $\mathcal{F}^{2k}(X)$ is generated by the elements $\{x_I\}_{I\in K}$ subject to the Minkowski relations

$$\sum_{I:|I|=k}a(I)\lambda(I)_\mu x_I=0,$$

where $\mu$ runs over $\Lambda^kV$ and $a$ runs over all functions such that the element

$$\sum_{I:|I|=k}a(I)[X_I/T]$$

bounds in $C_{n-k}(X/T;\mathbb{R})$.

This gives a combinatorial presentation of the vector-space structure of the part of cohomology generated by characteristic submanifolds. The source provides no resolution or further evidence for the claim.