The simple-matrix conjecture on linear homogeneous submatrices

Let $P$) be a simple zero-one matrix if both $P$ and its complement $P^c$ are acyclic, meaning that every submatrix has a row or column containing at most one 1-entry. A matrix is $P$-free if it contains no submatrix equal to $P$. A submatrix is homogeneous if all its entries are equal.

Simple-matrix conjecture. Let $P$ be a simple zero-one matrix. Then every $P$-free $n\times n$ zero-one matrix contains a $cn\times cn$ homogeneous submatrix, for a suitable constant $c>0$.

The preceding results establish this type of conclusion for the relevant $2\times k$ patterns, while nonsimple matrices admit counterexamples with only sublinear homogeneous submatrices. The conjecture concerns the remaining simple patterns, including the finite cases not covered by the proved results.