Gyárfás' staircase conjecture for binary matrices

Let $M$ be an $n\times n$ matrix whose entries are $0$ or $1$. A homogeneous staircase is a sequence of entries of one common value, with each successive entry either to the right in the same row or below in the same column; its size is its number of entries.

Gyárfás' conjecture. Every $0$-$1$ matrix $M$ of size $n\times n$ contains a homogeneous staircase of size at least $n-1$, equivalently, if $st(M)$ is the maximum size of a homogeneous staircase in $M$ and

$$st(n)=\min\{st(M):M\in\{0,1\}^{n\times n},\}$$

then $st(n)\geq n-1$. Since an example gives $st(n)\leq n-1$ for $n>1$, this is equivalently $st(n)=n-1$ for $n>1$.

This is a matrix formulation of a geometric Ramsey problem concerning monochromatic structures in convex, separated, balanced geometric complete bipartite graphs. The paper studies related symmetric and asymmetric extremal functions, but the supplied text gives no resolution of this conjecture.