GGC conjecture for large powers of positive stable variables

Let $Z_\alpha$ be a positive $\alpha$-stable random variable with $\alpha\in(0,1)$, and let $\gamma\in\mathbb{R}$. A positive random variable is generalized gamma convolution (GGC) if its Laplace transform has the representation specified in the paper. Stable-power GGC conjecture. The random variable

$$Z_\alpha^\gamma$$

is GGC for every $\alpha\in(0,1)$ satisfying

$$\lvert\gamma\rvert\ge\frac{\alpha}{1-\alpha}.$$

The conjecture is motivated by the paper's results on negative stable powers and by the implication that a positive answer to the conjecture for $V_\alpha^s$ would establish part of the positive-power range; it remains unproved in the supplied discussion.