Subsystems of the Walsh orthogonal system whose multiplicative completions are quasibases for $L^{p}[0,1]$, $1\leq p < +\infty$

If one discards some of the elements from the Walsh family, an ancient example of a system that serves as a Schauder basis for each of the $L^{p}$-spaces, with $1< p<+\infty$, the residual system will not be a Schauder basis for any of those spaces. Nevertheless, Price has shown that each member of a large class of such subsystems is complete on subsets of $[0,1]$ that have measure arbitrarily close to $1$. In the present work, it is shown that subsystems of this kind can be multiplicatively completed in such a way that the resulting systems are quasibases for each space $L^{p}[0,1]$, $1\le p<+\infty$, from which the earlier completeness result follows as a corollary.