The parameter-boundary conjecture for the reverse Mittag–Leffler inequality

Let $E_{\alpha,\beta}$ be the two-parameter Mittag–Leffler function. A complete Bernstein function is understood in the standard sense, and consider the global reverse inequality

$$\left|E_{\alpha,\beta}(z)\right|\ge E_{\alpha,\beta}(\operatorname{Re}z)\qquad\text{for every }z\in\mathbb{C}.$$

The parameter-boundary conjecture. There exists an increasing convex function $f:[2,\infty)\to[3,\infty)$ with $f(2)=3$ such that, for every $\alpha\ge2$,

$$\left|E_{\alpha,\beta}(z)\right|\ge E_{\alpha,\beta}(\operatorname{Re}z)\ \text{globally} \quad\Longleftrightarrow\quad \log E_{\alpha,\beta}\ \text{is a complete Bernstein function} \quad\Longleftrightarrow\quad \beta\le f(\alpha).$$

At $\alpha=2$ the corresponding boundary is known to be $f(2)=3$, and the paper gives partial support for extending this description to $\alpha>2$. The conjecture also predicts monotonicity and convexity of the boundary separating the parameter regions.