The dimension-three criterion for infinitely many H-trivial line bundles on toric DM stacks

Let $\mathbb{P}_{\mathbf{\Sigma}}$ be the smooth toric Deligne–Mumford stack associated to a complete stacky fan $\mathbf{\Sigma}=(\Sigma,\{v_i\}_{i=1}^{n})$ in a lattice of rank $3$. A line bundle is $\mathrm{H}$-trivial if all of its cohomology groups vanish. The vectors $v_i$ are the stacky fan generators, and a plane intersects a three-dimensional cone at its boundary when its intersection with that cone is contained in the cone's boundary.

The dimension-three H-triviality conjecture. There are infinitely many $\mathrm{H}$-trivial line bundles on $\mathbb{P}_{\mathbf{\Sigma}}$ if and only if there exists $\{i,j\}\subset\{1,2,\cdots,n\}$ such that $v_i$ and $v_j$ are collinear or there exists a plane intersecting all three-dimensional cones of $\mathbf{\Sigma}$ at their boundaries.

The conjecture is intended to encode the existence of fibrations producing infinitely many line bundles with vanishing cohomology in dimension three. The supplied text presents it as an expectation and gives no evidence that it has been proved or disproved.