On the approximation by the accompanying laws of $n$-fold convolutions of distributions with nonnegative characteristic functions

Let $F$ be a probability distribution on $R$ having nonnegative characteristic function and let $E$ be the distribution with the unit mass at the origin. It is proved that
$$ \sup_x|F^n([x,x+h))-e^{n(F-E)}([x,x+h))| \le C\gamma_h^{1/3}(|{\ln\gamma_h}|+1)^{13/3}n^{-1} $$
for any natural number $n$ and $h>0$. Here $C$ is an absolute constant and $\gamma_h$ denotes the value of the concentration function of the distribution $e^{n(F-E)}$ at the point $h$.