Complete monotonicity or log-convexity of a Mittag-Leffler expression

Let $0<\alpha\leq\frac{1}{2}$ and define

$$f(t)=\left|E_{2\alpha,1}(-t)\right|^2+t\left|E_{2\alpha,1+\alpha}(-t)\right|^2.$$

Here $E_{\rho,\beta}$ denotes the Mittag-Leffler function with parameters $\rho$ and $\beta$. Complete-monotonicity conjecture. The function $f(t)$ is completely monotone, or at least log-convex, for $t>0$. This property is proposed as the key ingredient for proving a logarithmic convexity estimate for time-fractional Schrödinger equations; no resolution is supplied, so the conjecture remains open.