Centralizer-centre conjecture for strange nilpotent orbits

Let $G$ be an algebraic group with Lie algebra ${\mathfrak g}$, let ${\mathcal N}$ be the nilpotent cone, and let $e\in{\mathcal N}$. Write ${\mathfrak g}^e$ for the centraliser of $e$ and ${\mathfrak z}({\mathfrak g}^e)$ for its centre. Centralizer-centre conjecture. If $G\cdot e\subset{\mathcal N}$ is strange and $e'\in{\mathfrak z}({\mathfrak g}^e)$, then $G\cdot e'$ is strange as well.

The conjecture is intended primarily for non-spherical orbits, since the centre of the centraliser of a spherical orbit is just $\Bbbk e$. Its truth would provide information about strange orbits beyond the spherical case.