Let $k$ be an algebraically closed field with $\mathrm{char}(k)=p>0$. Let $U$ be a connected unipotent algebraic group over $k$.

**Question:** When $p$ is big enough, is it true that $Z_U(u)$ is connected for any $u\in U$, or at least $u\in Z_U(u)^o$ for any $u\in U$?

**Remark:** This is true when $U$ is the unipotent radical of a Borel subgroup of a reductive group with $p$ being good (so that a Springer homeomorphism exists).