On narrow domains of the integral normal convergence

Let $X_{n1},\dots,X_{nk_n}$ be independently distributed random values with distribution functions $F_{n1}(x),\dots,F_{nk_n}(x)$, $n=1,2,\dots$. Let $c>0$ and let
$$ \int_{-\infty}^\infty x\,dF_{ni}(x)=0,\quad \int_{-\infty}^\infty x^2\,dF_{ni}(x)=\sigma^2_{ni}<\infty. $$
Put
$$ B_n^2=\sum_{i=1}^n\sigma_{ni}^2,\quad\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2}\,dt,\quad\log_mz=\underbrace{\log\log\dots\log}_{m\text{ раз}}z,\quad m\ge1. $$
Sufficient conditions are found for the relations to hold
\begin{gather} P\biggl(\frac{X_{n1}+\dots+X_{nk_n}}{B_n}\ge x\biggr)=(1-\Phi(x))(1+0(1)), \quad n\to\infty,\notag\\ P\biggl(\frac{X_{n1}+\dots+X_{nk_n}}{B_n}<-x\biggr)=\Phi(x)(1+0(1)), \quad n\to\infty, \notag \end{gather}
univormly in $x\in[0,c\sqrt{\log_mB_n^2}]$, $m\ge1$.