I tried to prove the following inequality:
For $\tau\ge 1$ and $A>0$, $$\frac{1}{\tau} \exp\left(A\right)E_1\left(A\right)\le \exp\left({\tau A}\right)E_1\left({\tau A}\right),$$ where $E_1(x)=\int_x^\infty \frac{\exp(-t)}{t} dt$ denotes the exponential integral.