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, whenever an entry $B_{i,j}$ is equal to $0$, **all** the entries of the $i$-th row **or** the $j$-th column of $B$ are equal to $0$ too.  

Namely, given any such block $B\in \{0,1\}^{r_B\times c_B}$ of $M$, $B_{i,j}=0$ implies ***(i)*** $B_{i,p}=0~~\forall p\in [c_B]$ **or** ***(ii)*** $B_{q,j}=0~~\forall q\in [r_B]$  (hence, we may even have simultaneously both ***(i)*** and ***(ii)***).

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

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