Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns.

Consider $\mathscr{M}[m^\sigma]$ to be collection of all $n\times n$ matrices obtained from matrices in $m^\sigma$ by swapping an equal number of rows for an equal number of columns of same indices.

As an example, say you pick row/column indices $i$ and $j$. Then you include matrix where you first replace $i$th row with transpose of $i$th column and vice versa followed by similar operation on $j$th row and column.

What is worst case difference between least rank and largest rank of any matrix in $\mathscr{M}[m^\sigma]$?

Can we say anything about their ratios (such as bound based on some intrinsic property of the matrix)?

I am guessing there is a constant $c\in[1,4]$ such that ratio of largest rank to least rank is bounded above by $c$. My guess is $c=4$.