Let $G$ be a simple group of lie type over a field of odd characteristic $p$, and $u$ is a regular unipotent element of $G$, that is a $p$-element with centralizer of minimum rank.
We know that centralizer of a regular unipotent element has to be a $p$-group.
We also know that every two regular unipotent elements of $G$ are conjugate, so there is only one conjugacy class of regular unipotent elements in $G$.
But what else we can say? Is it true that size of centralizer of a regular unipotent element divides size of centralizer of other unipotent elements?