Inequalities for the probabilities of large deviations in terms of pseudo-moments

Let $X_1,\dots,X_n$ be independent random variables with finite moments of order $t>2$ with zero means. Denote
\begin{gather*} \sigma_i^2=\mathbf EX_i^2,\quad c_{i,t}=\mathbf E|X_i|^t,\quad\sigma^2=\sum_{i=1}^n\sigma_i^2,\quad c_t=\sum_{i=1}^nc_{i,t},\quad L_t=c_t/\sigma^t, \\ S_n=\sum_{i=1}^nX_i. \end{gather*}
In [1] it was proved that
$$ \mathbf P(S_n\ge x\sigma)\le\exp(-K_1x^2)+K_2L_t/x^t $$
where $K_1$ and $K_2$ are constants dependent on $t$.
Our aim is to obtain an analogous inequality the right-hand side of which contains the so-called pseudo-moments $\nu_t$ instead of $c_{i,t}$, the pseudo-moments of a distribution $F(x)$ being defined as
$$ \nu_t(F)=t\int_{-\infty}^\infty|F(x)-\Phi_X(x)||x|^{t-1}\,dx $$
where $\Phi_X(x)$ is the normal distribution function with the same mean and variance as $F(x)$.