Ordered binary matrix removal lemma

Let an ordered binary matrix be a binary matrix whose rows and columns are equipped with orders. For a finite family $\mathcal{F}$ of ordered binary matrices, say that an $n \times n$ binary matrix is $\epsilon$-far from $\mathcal{F}$-freeness if changing fewer than an $\epsilon$-fraction of its entries cannot make it free of every matrix in $\mathcal{F}$ as an ordered submatrix. If a member of $\mathcal{F}$ has dimensions $a \times b$, let a copy mean an order-preserving occurrence of that matrix.

Ordered binary matrix removal lemma. For any finite family $\mathcal{F}$ of ordered binary matrices and any $\epsilon > 0$ there exists $\delta = \delta(\mathcal{F}, \epsilon)$ such that any $n \times n$ binary matrix which is $\epsilon$-far from $\mathcal{F}$-freeness contains at least $\delta n^{a+b}$ copies of some $a \times b$ matrix from $\mathcal{F}$.

This conjecture asks whether the efficient conditional regularity lemma for ordered binary matrices yields a removal lemma analogous to the unordered binary-matrix result. The supplied text attributes the conjecture to Alon, Fischer and Newman; its resolution is not specified here.