A hyperbolic lower bound for the Kilbas–Saigo function at the boundary parameter

Let $\alpha\in(0,1]$, $m>0$, and $x\geq 0$. Define the Kilbas–Saigo function $E_{\alpha,m,l}$ by its usual power series, and let $G$ denote the Barnes-type function used in the source. The conjectured lower bound. One has

$$E_{\alpha,m,m-\frac{1}{\alpha}}(-x)\geq \frac{1}{\left(1+(\alpha m)^{-\frac{\alpha}{m+1}}\left({\Gamma}(1+\alpha)G(1-\alpha;\alpha m)G(1+\alpha;\alpha m)\right)^{-\frac{m}{m+1}}x\right)^{1+\frac{1}{m}}}.$$

This is expected to complement the preceding uniform upper bound at the boundary value $l=m-1/\alpha$; the source indicates that the constant is suggested by the asymptotic behaviour of the associated density, but does not establish the inequality.