Let $A\in\mathbb{R}^{N\times N}$, and suppose that all elements of $A$ are distinct and non-zero. Further suppose that $A$ is full rank. Indeed, you should suppose that $A$ has any property held by almost all matrices (under the usual measure).
By way of background, note that by the algorithm given in Ruiz (2001), one can find diagonal matrices $R,C\in\mathbb{R}^{N\times N}$ and a matrix $B\in\mathbb{R}^{N\times N}$ with all rows and columns having sup norm $1$, such that $A=RBC$.
My question is the following, can one find a permutation matrix $P\in{\{0,1\}}^{N\times N}$, diagonal matrices $R,C\in\mathbb{R}^{N\times N}$ and a matrix $B\in\mathbb{R}^{N\times N}$ with all rows and columns having sup norm $1$, and with a unit diagonal, such that $A=RBCP$?
Numerical experiments suggest that if a counter-example exists, it has $N>2$. For $N=3$, counter-examples seem most likely in the form $$A=\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{bmatrix}+E,$$ where $E\in\mathbb{R}^{3\times 3}$ has small elements. However, it is far from immediate, since one can e.g. scale the middle row and column until the central element is equal to one, then scale down the first column and the last row so that the column/row's central element is at most one, then scale up the first row and the last column so that the diagonal is all ones.