String theory predicts the existence of extremely compact objects spinning faster than Kerr black holes. The spacetime exterior to such superspinars is described by Kerr naked singularity geometry breaking the black-hole limit on the internal angular momentum. We demonstrate that the conversion of Kerr superspinars into a near-extreme black hole due to an accretion counterrotating Keplerian disc is much more effective in comparison with the case of a corotating one since both the accreted rest mass necessary for conversion and the evolution time of conversion are by orders smaller for counterrotating discs. The conversion time of Kerr superspinars is given for several accretion regimes, and it is shown that the self-regulated accretion flow implies fastest evolution to the black-hole state. In the final stages of the conversion, Kerr superspinars can serve as very efficient particle accelerators in the region where the black-hole horizon forms.