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Abstract: Hybrid stars containing a quark matter core and a hadronic mantle may contain a sharp interface if the surface tension of dense matter is sufficiently high. When such a star is perturbed, conversion reactions may occur around the quark-hadron splitting surface. We show that the reaction speed has strong effects on the quasinormal mode spectrum and on stellar stability. In particular, if the conversion timescale is much larger than the perturbation one (slow reactions), changes of stellar stability do occur at critical points of the M(R) curve. As a consequence, a new class of dynamically stable hybrid stars is possible. We show that densities tens of times larger than the nuclear saturation density could be attained at the center of these hypothetic objects. We discuss possible formation mechanisms for the new class of hybrid configurations and smoking guns for their observational identification.