Complete-monotonicity criterion for the generalized Mittag-Leffler function

For $\alpha,\beta,\gamma>0$, let $F_{\alpha,\beta}^{(\gamma)}$ denote the generalized Mittag-Leffler function, and let $x>0$. Complete-monotonicity criterion conjecture. For every $\alpha,\beta,\gamma>0$, one has

$$F_{\alpha,\beta}^{(\gamma)}(-x)\;\text{is completely monotone}\quad\Longleftrightarrow\quad z+\gamma\bigl(z^{\beta-\alpha}-z^\beta\bigr)\leq 1\quad\text{for all }z\in(0,1).$$

The criterion is known to be sufficient, and its necessity was established when $\gamma\leq 1$ or when $\gamma>1$ and $\alpha\gamma=1$; the conjecture asserts necessity in all remaining cases.