Zero-free conjecture for the Mittag-Leffler function on the imaginary axis

Let $calpha a\in\left(\frac{3}{5}, 1\right)$. The Mittag-Leffler function $E_{\alpha,1}(z)$ is defined by its standard power series, and the imaginary axis is the set $\mathcal{\mathrm{i}t:t\in\mathbb{R}}$. Zero-free conjecture. The function $E_{\alpha,1}(z)$ has no zeros on the imaginary axis, namely

$$E_{\alpha,1}(\mathrm{i} t)\neq 0$$

for all $t\in\mathbb{R}$. This would extend the known result for $0<\alpha\leq\frac{3}{5}$ and would imply well-posedness of the corresponding backward fractional Schrödinger problem for all $T>0$ when $\nu=1$ and $0<\alpha<1$; the claim is motivated by numerical experiments and remains open.