The H-trivial line-bundle and piecewise-linear-function equivalence for toric DM stacks

For $\min \mathbb{N}$, let $\mathbb{P}_{\mathbf{\Sigma}}$ be a proper smooth dimension $m$ toric DM stack associated to a complete stacky fan $\mathbf{\Sigma}=(\Sigma, \{v_i\}_{i=1}^{n})$. An H-trivial line bundle means a line bundle satisfying the paper's H-triviality condition, and $\Lambda_{\psi}$ denotes the object associated to a $\mathbf{\Sigma}$-piecewise linear function $\psi$.

The H-trivial line-bundle and piecewise-linear-function conjecture. There are infinitely many H-trivial line bundles on $\mathbb{P}_{\mathbf{\Sigma}}$ if and only if there is a $\mathbf{\Sigma}$-piecewise linear function $\psi$ such that

$$\operatorname{dim}(\Lambda_{\psi})<m.$$

The paper proves the implication from the piecewise-linear condition to infinitely many H-trivial line bundles in full generality, and proves the converse when $m=3$ and there are no more than one collinear pair of rays. The equivalence in arbitrary dimension and without a bound on collinear ray pairs remains open.