Asymptotics of immaculate line bundles on smooth toric Deligne-Mumford stacks

A line bundle is immaculate if its cohomology vanishes in every dimension. We give a criterion for when a smooth toric Deligne-Mumford stack has infinitely many immaculate line bundles. This answers positively a question of Borisov and Wang. As a byproduct, we describe the asymptotic behaviour of the collection of immaculate line bundles.