The acyclic-matrix conjecture for dense zero-submatrices

A zero-one matrix $P$ is acyclic if every submatrix of $P$ has a row or column containing at most one 1-entry. A matrix is $P$-free if it contains no submatrix equal to $P$.

Acyclic-matrix conjecture. For every acyclic zero-one matrix $P$ and every $\varepsilon>0$, there is a $\delta>0$ such that every $P$-free $n\times n$ zero-one matrix with at least $\varepsilon n^2$ 0-entries contains a $\delta n\times\delta n$ all-0 submatrix.

This is presented as a strengthening of the simple-matrix conjecture. The proved results use the density of zero-entries to obtain large homogeneous submatrices in several special cases, but the asserted statement for arbitrary acyclic forbidden matrices remains open.