Linear monochromatic submatrix conjecture for bounded randomized communication

Let $M$ be an $n\times n$ Boolean matrix, and let randomized communication complexity mean the public-coin randomized communication complexity of $M$. A submatrix is monochromatic if all its entries are equal, so an all-zeros or all-ones submatrix is monochromatic. Linear monochromatic-submatrix conjecture. Every Boolean matrix with randomized communication complexity bounded by a constant contains an all-zeros or all-ones submatrix of linear size.

This conjecture asks for a strong structured rectangle in matrices of constant randomized communication complexity. It remains open; the source presents it as a basic structural question related to approximate $\gamma_2$-norm and communication complexity.