HCM conjecture for square roots of products of gamma variables

Let $\gamma_t$ and $\gamma_s$ be independent gamma random variables with parameters $t,s>0$. The gamma-product conjecture. For all $s,t>0$, the independent product $\sqrt{\gamma_t\gamma_s}$ is $\operatorname{HCM}$.

The surrounding discussion proves the weaker range $|t-s|\le 1/2$ using a product formula for modified Bessel functions. The assertion for arbitrary positive $s$ and $t$ is presented as a natural conjectural extension and is not resolved in the supplied text.