Let $G$ be a semisimple, simply-connected algebraic group over an algebraically closed field $k$ of positive characteristic. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$. Also let $P$ be a parabolic subgroup of $G$ containing $B$ and let $L$ be its Levi factor. Denote by $U_P \subseteq U$ the unipotent radical of $P$ and set $U_L := U \cap L$.

Let $\mathfrak U \subseteq G$ denote the unipotent variety and let $ \mathfrak R \subseteq \textrm{Lie}(G) $ denote the nilpotent cone. If $p$ is a good prime for $G$ then there is a $G$-equivariant Springer isomorphism $\phi : \mathfrak U \to \mathfrak R$ that restricts to an isomorphism $ U \to \textrm{Lie}(U) $. My question is: Does $\phi$ restrict to isomorphisms $ U_P \to \textrm{Lie}(U_P) $ and $U_L \to \textrm{Lie}(U_L)$? If this does not always happen, are there conditions on the parabolic $P$ under which it will be true?