Not yet reviewed by Pith; the record is open.

One of the most frequently used indicators to characterize the magnetic field's influence on star formation is the relation between magnetic field strength and gas density ($B-\rho$ relation), usually expressed as $B \propto \rho^{\kappa}$. The value of $\kappa$ is an indication of the dynamical importance of the magnetic field during gas compression. Investigating the global magnetic field's impact on this relation and its evolution, we conduct MHD simulations of Milky-Way-like galaxies including gravity, star formation, and supernova feedback along with non-equilibrium chemistry up to $H_2$ formation fueling star formation. Two initial magnetic field morphologies are studied: one completely ordered (toroidal) and the other completely random. In these models, we study the dynamical importance of the magnetic field through the plasma $\beta$ and the $B-\rho$ relation. For both magnetic morphologies, low-density regions are thermally supported, while high-density regions are magnetically dominated. Equipartition is reached earlier and at lower densities in the toroidal model. However, the $B-\rho$ relation is not unique even within the same galaxy, as it consistently includes two different branches for a given density, with $\kappa$ ranging from about 0.2 to 0.8. The mean value of $\kappa$ for each model also displays significant variations over time, which supersede the differences between the two models. While our findings suggest that the magnetic field morphology does influence the galactic $B-\rho$ relation, its impact is transient, since time-averaged differences between the models fall within the large temporal scatter. The context and time-dependent nature of the $B-\rho$ relation underscore the need for comprehensive research and observations to understand the intricate role of magnetic fields in star formation processes across diverse galactic environments.