Virtual homological torsion: abundance versus growth in books of $I$-bundles

Let $\mathcal{B}$ be a book of $I$-bundles, all of whose pages are surfaces of negative Euler characteristic. In this short note, we prove that torsion in the first homology of $\mathcal{B}$ grows subexponentially in the index along any exhausting tower of regular finite-sheeted covers. By contrast, recent work of Ascari and the author shows that, apart from the obvious exceptions, $\mathcal{B}$ has abundant virtual homological torsion, which can grow exponentially along exhausting towers of non-regular finite covers.