Dimension dependence of factorization problems: bi-parameter Hardy spaces

Given $1 \leq p,q < \infty$ and $n\in\mathbb{N}_0$, let $H_n^p(H_n^q)$ denote the canonical finite-dimensional bi-parameter dyadic Hardy space. Let $(V_n : n\in\mathbb{N}_0)$ denote either $\bigl(H_n^p(H_n^q) : n\in\mathbb{N}_0\bigr)$ or $\bigl( (H_n^p(H_n^q))^* : n\in\mathbb{N}_0\bigr)$. We show that the identity operator on $V_n$ factors through any operator $T : V_N\to V_N$ which has large diagonal with respect to the Haar system, where $N$ depends \emph{linearly} on $n$.