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More than 20 years ago, Glendenning, Kettner and Weber proposed the existence of stable white dwarfs with a core of strange quark matter. More recently, by studying radial modes, Alford, Harris and Sachdeva concluded instead that those objects are unstable. We aim to clarify this issue. We investigate the stability of these objects by looking at their radial oscillations while incorporating boundary conditions at the quark-hadron interface, corresponding to either a rapid or a slow conversion of hadrons into quarks. Our analysis shows that objects of this type are stable if the star is not strongly perturbed, and ordinary matter cannot transform into strange quark matter because of the Coulomb barrier separating the two components. On the other hand, ordinary matter can be transformed into strange quark matter if the star undergoes a violent process, as in the preliminary stages of a type Ia supernova, and this causes the system to become unstable and to collapse into a strange quark star. In this way, accretion-induced collapse of strange dwarfs can be facilitated, and km-sized objects with subsolar masses can be produced.