If $A$ and $B$ have the same degree of nilpotence, do they have the same rank?

Question

Let $A, B$ be nilpotent $n\times n$ matrices over the field $K$. Is the following correct?

If $A$ and $B$ has the same degree of nilpotency, then $\operatorname{rank} A = \operatorname{rank} B $

Answer 1

This is not correct. As a counter-example, consider $$ A = \pmatrix{0&1\\&0\\&&0\\&&&0}, \quad B = \pmatrix{0&1\\&0\\&&0&1\\&&&0} $$ which both have nilpotency degree $2$.

This is true, however, for $n \leq 3$, since nilpotent matrix of the same nilpotency degree are necessarily similar (since they have the same Jordan form).