Unimprovability of the result due to N. A. Sapogov in the stability problem of Cramér's theorem

We consider the sequence (1) of compositions of distribution functions satisfying the condition (2). Let truncated variances of components be bounded from below by a positive constant. It is proved that the well-known estimate
$$ \max_{i=1,2}\inf_{G\in N}\sup_x|F_n^{(i)}(x)-G(x)|=O\biggl(\frac1{\sqrt{-\ln\varepsilon_n}}\biggr) $$
(where $N$ is the set of all normal distribution functions) is unimprovable.