Given two real and irreducible matrices $A$ and $B$ of size $n \times n$. A matrix $A$ is irreducible if there is no permutation matrix $Q$ so that $$ Q^{-1} A Q = \begin{bmatrix} E & G \\ 0 & F \end{bmatrix} $$ where $E$ and $F$ are square.

Also, $A$ and $B$ have equal principal minors, i.e, $$ det A(\alpha) = det B(\alpha) \textrm{ for non-empty } \alpha \subset \{1, \dots, n\}, $$ where $A(\alpha)$ is the submatrix with columns and rows indices in $\alpha$.

There is also symmetric $P$ such that $$ P A P^{-1} = B. $$ Are then $A$ and $B$ necessarily diagonally similar? The opposite direction is clear, i.e., for $A$ and $B$ being diagonally similar, $A$ and $B$ have equal principal minors.