Infinite divisibility of $\alpha$-Cauchy distributions

In 2009, Yano, Yano and Yor proposed the question of studying the infinite divisibility of the $\alpha$-Cauchy variable ${\mathcal{C}}_{\alpha }$ for $\alpha >1$. The particular case ${\mathcal{C}}_2$ is the well-known standard Cauchy variable, which is infinitely divisible and indeed stable. For $\alpha \mathrm{\ne }2$, the infinite divisibility of ${\mathcal{C}}_{\alpha }$ is previously unknown. In this paper, we prove that ${\mathcal{C}}_{\alpha }$ is infinitely divisible if and only if $1<\alpha \le 2$.