On the strengthened $L^1$-greedy property of the Walsh system

For any $0<\varepsilon<1$ a measurable set $E\subset[0,1]$ exists with a measure $|E|>1-\varepsilon$ such that for each function $f(x)\in L^1(0,1)$ one can find a function $g(x)\in L^1(0,1)$, which coincides with $f(x)$ on $E$, such that its Fourier–Walsh series converges to it in the $L^1(0,1)$-metrics, and all nonzero terms of the sequence of the Fourier coefficients of the new function obtained by the Walsh system have the modulo decreasing order, and, consequently, the greedy algorithm for this function converges to it in the $L^1(0,1)$-norm.