GGC conjecture for negative powers of Beta variables

Let $\operatorname{Beta}(\alpha_1,\alpha_2)$ denote a Beta random variable with parameters $\alpha_1,\alpha_2>0$, and let $s\ge1$. A positive random variable is generalized gamma convolution (GGC) if its Laplace transform has the representation specified in the paper. Beta-power GGC conjecture. The random variable

$$\bigl(\operatorname{Beta}(\alpha_1,\alpha_2)\bigr)^{-s}$$

is GGC for every $\alpha_1,\alpha_2>0$ and every $s\ge1$. The paper records only partial results, including the case $\operatorname{Beta}(1/2,1/2)$ for $s\in[1/2,1]$, and states that no general result on infinite divisibility of negative powers of Beta variables was known.