Let $A$ be an $n \times n$ **real** matrix with the following property:

All the conjugates of $A$ have only zeros on the diagonal. Does $A=0$?

(By conjugates, I mean all the matrices similar to it, over $\mathbb{R}$, that is I require the conjugating matrix to be real).

Of course, if $A$ is diagonalizable, then clearly it must be zero.

The only idea I have is to use the Jordan form for real matrices, but after some thought I am not sure this is a good approach.