  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.
