Does $C$ diagonal + nonnegative, $ACB^T$ diagonal + nonnegative imply $A =B$?

Question

If $C\in \mathbb R^{m,m}$ is a diagonal nonnegative matrix and $X = ACB^T\in \mathbb R^{n,n}$ is a diagonal nonnegative matrix, does this imply $A = B$? Is there a counterexample?

Answer 1

Take $A=C=I_m$ and $B=diag(1,2,\ldots,m)$.

Answer 2

For almost any $C$ you can find $A\neq B$ satisfying this.

Let $A$ be a permutation matrix, and $C$ a positive diagonal matrix. For any positive number $b$ define $B=bA$. Now check that $ACB^T$ is diagonal and positive and so provides plenty of counter examples.