If $\operatorname{rank}(A)=\operatorname{rank}(AB)$, prove that $\operatorname{rank}(AB)=\operatorname{rank}(ABA)$

Question

Let $A,B$ be $n\times n$ matrixes over a field $K$. If $\operatorname{rank}(A)=\operatorname{rank}(AB)$, prove that $\operatorname{rank}(AB)=\operatorname{rank}(ABA)$.

Answer 1

It's false, see for instance $A=\begin{bmatrix}0&0\\ 1&0\end{bmatrix}$, $B=\begin{bmatrix}1&0\\ 0&0\end{bmatrix}$.