We are given a matrix $M \in \{0,1\}^{n\times n}$ satisfying the following property.

The rows and columns of $M$ can be partitioned into $k$ rowgroups and $k$ colgroups respectively, such that in each block $B \subseteq M$ induced by these partitions, we have $0$ or more rows and $0$ or more columns of $B$ containing only entries equal to $0$, while all the remaining entries of $B$ are equal to $1$.

Question: What is the maximum rank of $M$?

I only know the maximum rank of $M$ is upper bounded by $k^2$.