Hyperbolic two-sided bounds for the Kilbas–Saigo function

Let $\alpha\in(0,1]$, $m>0$, $l>m-1/\alpha$, and $x\geq 0$. Let $E_{\alpha,m,l}$ be the Kilbas–Saigo function. The conjectured two-sided bound. One has

$$\frac{1}{1+\frac{{\Gamma}(1+\alpha(l-m))}{{\Gamma}(1+\alpha(l-m+1))}x}\leq E_{\alpha,m,l}(-x)\leq \frac{1}{1+\frac{{\Gamma}(1+\alpha l)}{{\Gamma}(1+\alpha(1+l))}x}.$$

The bounds are natural because the constants match the small- and large-$x$ asymptotic regimes. The source states that the conjecture is known when $l=m-1$ and when $m=1$, while the general case remains open.