The polynomial removal lemma conjecture for ordered binary matrices

A binary matrix is a matrix whose entries lie in $\{0,1\}$. A copy of a $k\times k$ binary matrix $A$ in an $n\times n$ binary matrix $M$ consists of increasing row indices $r_1<\dots<r_k$ and increasing column indices $c_1<\dots<c_k$ such that

$$M_{r_i,c_j}=A_{i,j}\qquad(1\leq i,j\leq k).$$

Ordered matrix removal conjecture. For every $k\times k$ binary matrix $A$ and every $\varepsilon>0$, there is a $\delta=\delta(k,\varepsilon)=\operatorname{poly}(\varepsilon)>0$ such that, whenever at least $\varepsilon n^2$ entries of an $n\times n$ binary matrix $M$ must be changed to eliminate all copies of $A$, the matrix $M$ contains at least

$$\delta n^{2k}$$

copies of $A$. This is a central open problem for ordered matrices; the source notes that it is not known even for the $2\times2$ identity matrix.