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Fast radio bursts (FRBs) are millisecond-duration extragalactic radio transients with extremely high brightness temperatures. Magnetars are widely considered leading candidates for their progenitors. In a few hyperactive repeating FRBs, a co-spatial, compact, and incoherent persistent radio source (PRS) has been detected, providing a powerful probe of the physical environment and energetics of the central engine. Radio imaging, scintillation measurements, and equipartition arguments all point to a very compact PRS, placing strong constraints on the surrounding medium. This compactness is difficult to reconcile with the presence of a rapidly expanding supernova remnant, as expected from a millisecond magnetar formed in a highly energetic explosion such as a superluminous supernova, despite such scenarios being widely favored in the literature. Instead, we argue that a sub-energetic supernova giving birth to a strongly magnetized neutron star with an initial spin period of tens of milliseconds can naturally account for the observed PRS properties. In addition, this scenario makes concrete predictions for the broadband PRS spectrum: a synchrotron self-absorption break near 200 MHz and a cooling break at around 150 GHz. Both of these spectral features can be tested with existing radio facilities, offering a direct observational pathway to constrain the nature of FRB progenitors.