Conjecture on the zeros of the Mittag–Leffler function with second parameter 2

Let $E_{\alpha,\beta}$ denote the Mittag–Leffler function. There exists some $\alpha_0\in(1.59911,1.59912)$, and for $\alpha\geq\alpha_0$ let $Z_2(\alpha)$ denote the smallest positive solution of $E_{\alpha,2}(-z^\alpha)=0$.

Smallest-zero conjecture for $E_{\alpha,2}$. For each $\alpha\in[\alpha_0,2]$, the equation

$$E_{\alpha,2}(-z^\alpha)=0$$

has at least one solution in $(0,\infty)$. As $\alpha\to2-$,

$$Z_2(\alpha)=\pi+(2-\alpha)+o(2-\alpha),$$

and as $\alpha\to\alpha_0+$,

$$Z_2(\alpha)=Z_2(\alpha_0)+c(\alpha-\alpha_0)^b(1+o(1)),$$

where $Z_2(\alpha_0)\approx5.21066$, $b\approx1/2$, and $c\approx-4.8$. Furthermore, $Z_2(\alpha)$ is strictly decreasing in $\alpha$. For each $\alpha\in(1,\alpha_0)$, $E_{\alpha,2}(-z^\alpha)>0$ for all $z\geq0$.

The threshold and the asserted behavior are based on numerical observations; the exact value of $\alpha_0$ and the stated asymptotics and monotonicity remain unproved here. The context attributes the existence of a threshold to earlier work but does not establish these precise claims.