On Taylor coefficients and $L_p$-continuity moduli of Blaschke products

Let
$$ B=\prod_{k\geq1}b_{z_k},\quad b_{z_k}\overset{\text{def}}=\frac{|z_k|}{z_k} \frac{z_k-z}{1-\bar z_k z},\quad |z_k|<1, $$
be a Blaschke product, let $\widehat{B}(z)$ denote its $k$-th Taylor coefficient. Suppose $\{z_k\}$ splits into a finite union of sequences $\{\xi_k\}$ satisfying
$$ \sup_{k\geq1}\frac{1-|\xi_{k+1}|}{1-|\xi_{k}|}<1. $$
The following assertions are proved to be equivalent:
1. $\{z_k\}_{k\geq1}\in(\omega N)$;
2. $\widehat{B}(k)=O(1/k)$, $k\to\infty$;
3. $\sum_{k\geq n}|\widehat{B}|^2=O(n^{-1})$;
4. $B\in\operatorname{Lip}(1/p,L^p)$ for some $p$, $1<p<\infty$.