On strengthening of Lyapunov type estimates (the case when summands distributions are close to the normal one)

Let $\{X_j\}_{j=1}^n$ be a sequence of independent random variables. Put
\begin{gather*} \mathbf MX_j=0,\quad\mathbf MX_j^2=\sigma_j^2,\quad B^2=\sum_{j=1}^n\sigma_j^2,\quad C=\sum_{j=1}^n\sigma_j^3; \\ \nu_j=3\int_{-\infty}^\infty x^2|F_j(x)-\Phi(x/\sigma_j)|\,dx \end{gather*}
where $F_j(x)=\mathbf P\{X_j<x\}$, $\Phi(x)=(2\pi)^{-1/2}\int_{-\infty}^xe^{u^2/2}\,du$. Let
$$ \Lambda=\sum_{j=1}^n\nu_j,\quad\delta=\sup_x\biggl|\mathbf P\biggl\{\sum_{j=1}^nX_j<Bx\biggr\}-\Phi(x)\biggr|. $$
In the paper, some estimates of $\delta$ are obtained. The simpliest consequence from these estimates is the following:
$$ \delta\le L\max\biggl\{\frac\Lambda{B^3};\biggl(\frac{\Lambda}{B^3}\biggr)^{1/4}\biggl(\frac C{B^3}\biggr)^{3/4}\biggr\} $$
where $L$ is an absolute constant.