On quasi-universal Walsh series in $L^p_{[0,1]}$, $p\in[1,2]$

Let the sequence $\{a_{k}\}_{k=1}^{\infty},$ $a_{k}\searrow0$ with $\{a_{k}\}_{k=1}^{\infty}\notin l_{2},$ and Walsh system $\{W_{k}(x)\}_{k=0}^{\infty}$ be given. Then for any $\epsilon>0$ there exists a measurable set $E\subset\lbrack0,1]$ with measure $|E|>1-\epsilon$ and numbers $\delta_{k}=\pm1, 0$ such that for any $p\in\lbrack1,2]$ and each function $f(x)\in L^{p}(E)$ there exists a rearrangement $k\to\sigma(k)$ such that the series $\displaystyle\sum _{k=1}^{\infty}\delta_{\sigma(k)}a_{\sigma(k)}W_{\sigma(k)}(x)$ converges to $f(x)$ in the norm of $L^{p}(E)$.