The acyclic forbidden-matrix conjecture for homogeneous submatrices

A zero-one matrix $P$ is acyclic if every submatrix of $P$ has a row or column containing at most one 1-entry; its complement $P^c$ is obtained by exchanging 0-entries and 1-entries. A matrix is $P$-free if it contains no submatrix equal to $P$, and a submatrix is homogeneous if all entries are equal.

Acyclic forbidden-matrix conjecture. Let $P$ be an acyclic zero-one matrix. Then every $n\times n$ zero-one matrix that is both $P$-free and $P^c$-free contains a $cn\times cn$ homogeneous submatrix, for a suitable constant $c>0$.

The statement is an immediate corollary proposed from the preceding density conjecture, by applying it to a matrix or its complement. It extends the known linear homogeneous-submatrix results beyond simple matrices and remains open.