The estimates of stability of the characterization of the normal distribution given by G. Polya theorem

Let $X_1$, $EX_1=0$ and $X_2$ be independent identically distributed random variables and let
$$ \sup_x|P(X_1<x)-P(aX_1+bX_2<x)|\leq\varepsilon, $$
where $a>0$, $b>0$, $a^2+b^2=1$. Suppose that for some integer $r\geq3p$, $P=(1/2\ln(1/2))/\ln(\max(a,b))$,
$$ \mathsf{E}|X_1|^r\leq M_r<\infty;\quad\mathsf{E}X_i^s=\mathsf{E}N_{0,\sigma}^s,\quad s=1,2,\dots,r-1, $$
where $N_{0,\sigma}$ – normal random variable with parameters $(0,\sigma)$. Then exist such constants $c=c(b,M_r)$ and $\varepsilon_0=\varepsilon_0(b,M_r,\sigma)$ that
$$ \sup_x|P(X_1<x)-\frac1{\sigma\sqrt{2\pi}}\int_{-\infty}^x e^{-y^2/2\sigma^2}\, dy|\leq c(\sigma^{-2p}+\sigma^{r-2p})\varepsilon^{1-2p/r}. $$