Dimension dependence of factorization problems: Hardy spaces and $SL_n^\infty$

Given $1 \leq p < \infty$, let $W_n$ denote the finite-dimensional dyadic Hardy space $H_n^p$, its dual or $SL_n^\infty$. 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 $n$.