HCM Property and the Half-Cauchy Distribution

Let $Z_\al$ be a positive $\alpha$-stable random variable and $T_\al=(Z_\al/\tilde Z_\al)^\al,$ with independents components in the quotient. It is known that $T_\al$ is distributed as the positive branch of a Cauchy random variable with drift. We show that the density of the power transformation $T_\al^\beta$ is hyperbolically completely monotone in the sense of Thorin and Bondesson if and only if $\al\le1/2$ and $|\beta|\ge 1/(1-\al).$ This clarifies a conjecture of Bondesson (1992) on positive stable densities.