Metrization of powers of the Jensen-Shannon divergence

Metrization of statistical divergences is valuable in both theoretical and practical aspects. One approach to obtaining metrics associated with divergences is to consider their fractional powers. Motivated by this idea, Osán, Bussandri, and Lamberti (2018) studied the metrization of fractional powers of the Jensen-Shannon divergence between multinomial distributions and posed an open problem. In this short note, we provide an affirmative answer to their conjecture. Moreover, our method is also applicable to fractional powers of $f$-divergences between Cauchy distributions.