On the stability of decompositions of the unit distribution function

Let $E$ be the unit distribution function
$$ E(x)= \begin{cases} 0,&x\le0 \\ 1,&x>0 \end{cases} $$
and $F=F_1*F_2$ be a distribution function such that in the uniform metric
$$ \rho(F,E)\le\varepsilon\le1/4. $$
Let $F_1$ have median 0. We show that
$$ \rho(F_1,E)\le\frac{1-\sqrt{1-4\varepsilon}}2. $$
and this estimate can not be improved.