For a natural number $n$, let $Y_n$ denote the linear span of the first $n+1$ levels of the Haar system in a Haar system Hardy space (i.e., in a rearrangement-invariant function space or a related space such as dyadic $H^1$). Consider the following question: Given $n$ and $\delta > 0$, how large does $N$ have to be chosen such that for every linear operator $T$ on $Y_N$ with norm at most 1 and with $\delta$-large positive diagonal entries (with respect to the Haar basis), there exists a factorization of the identity $I_{Y_n} = ATB$, where $A$ and $B$ satisfy a uniform upper bound such as $\|A\|\|B\|\leq 2$? This problem is closely related to the Restricted Invertibility Theorem by J. Bourgain and L. Tzafriri. We show that in general, an inequality of the form $N \ge Cn^2$ is sufficient, whereas if the Haar system is unconditional in the underlying space, then $N\ge Cn$ suffices. This amounts to a quasi-polynomial or, respectively, polynomial dependence between the dimensions of $Y_N$ and $Y_n$.