Convexity conjecture for the Gamma-type non-negativity domains

For positive real numbers $a$ and $b$, let ${\mathcal{D}}_{a,b}$ denote the parameter set introduced in the paper for the corresponding Gamma-type non-negativity problem.

Convexity conjecture. For every $a,b>0$, the set ${\mathcal{D}}_{a,b}$ is convex.

This is posed as a natural open problem arising from the probabilistic reformulation of non-negativity questions for hypergeometric functions and Bessel functions of the first kind; no resolution is given here.