Borel subgroups of centralisers of Lie algebra elements in bad characteristic Let $G$ be a simple linear algebraic group over an algebraically
closed field $k$ of characteristic $p>0$, and let $\mathfrak{g}=\mathrm{Lie}(G)(k)$
denote (the $k$-points of) the Lie algebra. 

Question. Assume that $p$ is not very good for $G$. For
  $X\in\mathfrak{g}$, is there a Borel subgroup $B_{0}\subseteq
C_{G}(X)_{\mathrm{red}}^{\circ}$ and a Borel subgroup $B\subseteq G$
  such that $X\in\mathrm{Lie}(B)(k)$ and $B_{0}\subseteq B$? 
  (Here
  $C_{G}(X)_{\mathrm{red}}^{\circ}$ denotes the connected component of
  the reduced subgroup of the centraliser $C_{G}(X)$.)

If $p$ is very good for $G$ it is known that $C_{G}(X)$ is reduced,
so $C_{\mathfrak{g}}(X)=\mathrm{Lie}(C_{G}(X))(k)$, and hence $X\in\mathrm{Lie}(C_{G}(X))(k)$.
By a theorem of Grothendieck (see 14.25 in Borel's book on linear
algebraic groups), the Lie algebra of any smooth algebraic group is
covered by Borel subalgebras (i.e., Lie algebras of Borel subgroups),
so there exists a Borel subgroup $B_{0}\subseteq C_{G}(X)^{\circ}$ such that
$X\in\mathrm{Lie}(B_{0})(k)$. Since $B_{0}$ lies in some Borel subgroup
of $G$, the desired conclusion follows.
When $p$ is not very good for $G$, it is not true in general that
there is a Borel $B_{0}\subseteq C_{G}(X)_{\mathrm{red}}^{\circ}$ such that
$X\in\mathrm{Lie}(B_{0})(k)$. For example, take $G=\mathrm{SL}_{p}$
and $X=\lambda+Y$, where $\lambda$ is any non-zero scalar matrix
and $Y$ is regular nilpotent. However, the above question has a positive
answer in this case since $C_{G}(X)_{\mathrm{red}}^{\circ}$ is its own Borel and $C_{\mathfrak{g}}(X)=\mathrm{Lie}(C_{G}(X)_{\mathrm{red}})(k)\oplus\mathfrak{z}$ (where $\mathfrak{z}$ is the centre of $\mathfrak{g}$), which embeds
in the Lie algebra of a Borel of $G$. 
Added. Since the question can (in principle) be decided by going through the nilpotent $X$, I would also be interested in references to sources which may be helpful in carrying out the case by case analysis (i.e., containing explicit information about the Borels in $C_G(X)_{\mathrm{red}}^{\circ}$ and the structure of $C_{\mathfrak{g}}(X)$).
 A: I think the positive answer to this question follows from some results obtained in the paper The Hesselink stratification of nullcones and base change, Invent. Math., 191 (2013), 631-669, by M. Clarke and A. Premet. It is proved in the paper among other things that for any nonzero nilpotent element $e\in \mathfrak{g}$ there exists an optimal cocharacter $\lambda\in X_*(G)$ for $e$ such that $e\in \bigoplus_{i\ge 2}\,\mathfrak{g}(\lambda,i)$ and $C_G(e)_{\rm red}\subseteq P(\lambda)$, where $P(\lambda)$ is the parabolic subgroup of $G$ with Lie algebra $\bigoplus_{i\ge 0}\,\mathfrak{g}(\lambda,i)$. If we choose $e\in\mathfrak{g}$ nicely then this cocharacter is obtained by base-changing one of the Dynkin cocharacters (defined over the integers). Any Borel subgroup $B$ of $P(\lambda)$ has that property that $\bigoplus_{i\ge 1}\,\mathfrak{g}(\lambda,i)\subset {\rm Lie}(B)$ and hence ${\rm Lie}(B)$ contains $e$. So any $B$ as above containing a Borel subgroup of $C_G(e)_{\rm red}\subset P(\lambda)$ would do.
