On existence of a universal function for $L^p[0,1]$ with $p\in(0,1)$

We show that, for every number $p\in(0,1)$, there is $g\in L^1[0,1]$ (a universal function) that has monotone coefficients $c_k(g)$ and the Fourier–Walsh series convergent to $g$ (in the norm of $L^1[0,1]$) such that, for every $f\in L^p[0,1]$, there are numbers $\delta_k=\pm1,0$ and an increasing sequence of positive integers $N_q$ such that the series $\sum^{+\infty}_{k=0}\delta_kc_k(g)W_k$ (with $\{W_k\}$ the Walsh system) and the subsequence $\sigma^{(\alpha)}_{N_q}$, $\alpha\in(-1,0)$, of its Cesáro means converge to $f$ in the metric of $L^p[0,1]$.