Metrization of powers of the Jensen-Shannon divergence

Metrization of statistical divergences is useful in both theoretical and practical aspects. Considering the fractional powers of statistical divergences is one way to obtain metrics associated with divergences. With this motivation, Os\'an, Bussandri, and Lamberti (2018) considered metrizations for the fractional powers of the Jensen-Shannon divergences between multinomial distributions and gave an open problem. In this short note, we give an affirmative answer to their conjecture. This method is also applicable to powers of f-divergences between the Cauchy distribution.