The large empty bipartite pair conjecture under induced four-cycle-free restrictions

Let $G$ be a bipartite graph with vertex classes $A$ and $B$, each of size $n$. The maximum degree of $G$ is at most $\varepsilon n$, and an induced four cycle-free subgraph is an induced subgraph containing no cycle of length four. The large empty bipartite pair conjecture. For every $k>0$ there exist $\varepsilon,c>0$ such that, if $G$ contains no induced four cycle-free subgraph of average degree more than $k$, then there exist $X\subset A$ and $Y\subset B$ with $|X|=|Y|\geq cn$ and no edges between $X$ and $Y$. The source states that proving this combinatorial assertion would suffice for the large constant submatrix consequence of the preceding conjecture, but gives no resolution status.