Generalized gamma convolution threshold conjecture for ${\bf K}_\alpha$

Let ${\bf K}_\alpha$ be the random variable defined in the paper, with parameter $\alpha\in(0,1)$. A positive random variable has a generalized gamma convolution (GGC) law if its distribution belongs to the class of generalized gamma convolutions. GGC threshold conjecture. The law of ${\bf K}_\alpha$ is a GGC if and only if

$$\alpha\leq \frac{4}{5}.$$

The conjecture is motivated by the paper's analysis of the monotonicity of the function $\theta_a$ and gives a sharp proposed boundary for the GGC property; no resolution is supplied in the excerpt.