Let $G$ be a connected linear algebraic group over an algebraically
closed field $k$ of characteristic $p$. An element $x\in G$ is
called *regular* if its centralizer has minimal dimension among
all the elements of $G$. Suppose now that $G$ is connected simple,
let $u\in G$ be a regular unipotent element, and let $U$ be the
unipotent radical of the Borel containing $u$. Then Springer [Some
arithmetical results on semi-simple Lie algebras] has shown that
the centralizer $C_{U}(u)$ is connected, provided $p$ is a good
prime for $G$.

Is it true that if $p$ is a good prime for $G$, then $C_{U}(x)$ is connected for all $x\in U$?

Let $V$ by an arbitrary connected unipotent group over $k$. Is it possible to describe some 'small' subset $S$ of $V$, such that the connectedness of $C_{V}(x)$ for all $x\in S$ implies the connectedness of $C_{V}(x)$ for all $x\in V$? In particular, can $S$ be taken to be the set of regular elements in $V$?