On unconditional and absolute convergence of the Haar series in the metric of $L^{p}[0,1]$ with $0<p<1$

We prove that there exists a universal Haar series of the form $\sum\nolimits_{k=0}^{\infty}a_{k}h_{k}(x)$ with $a_{k}\searrow 0$ such that, for all $0<p<1$ and $f\in L^{p}[0,1)$, there is a numeric sequence $\{\delta_{k}:\delta_{k}=1$ or $0$, $k=0,1,2,\dots \}$, for which the series $\sum\nolimits_{k=0}^{\infty}\delta_{k}a_{k}h_{k}(x)$ converges absolutely to $f(x)$ in $ L^{p}[0,1)$.