Let $G= \mathbb{G}_m^k$ act on a variety $X$.
Let $\mathcal{L}$ be a line bundle on $X$ and assume that for each $g \in G$ the pullback $g^\star \mathcal{L}$ is isomorphic to $\mathcal{L}$.
Does it necessarily follow that $\mathcal{L}$ can be pulled back from a line bundle on the quotient stack $[X/\mathbb{G}_m^k]$?
I am particularly interested in the case that $X$ is obtained from several toric varieties by identifying boundary divisors.
