Dimension dependence of factorization problems: Hardy spaces and $S{L}_n^{\mathrm{\infty }}$

Given $1\le p<\mathrm{\infty }$, let $W_n$ denote the finite-dimensional dyadic Hardy space $H_n^p$, its dual or $S{L}_n^{\mathrm{\infty }}$. We prove the following quantitative result: The identity operator on $W_n$ factors through any operator $T:W_N\to W_N$ which has large diagonal with respect to the Haar system, where $N$ depends \emph{linearly} on $n$.