Power conjecture for strange nilpotent orbits in ${\mathfrak{sl}}_n$

Let ${\mathcal N}\subset{\mathfrak{sl}}_n$ be the nilpotent cone, let $e\in{\mathcal N}$, and define ${\mathcal O}^{\langle k\rangle}:=SL_n\cdot e^k$ for $k\geqslant1$. Power conjecture. If ${\mathcal O}=SL_n\cdot e\subset{\mathcal N}$ is strange, then ${\mathcal O}^{\langle k\rangle}$ is strange for every $k\geqslant1$.

This is presented as the specialization to ${\mathfrak{sl}}_n$ of the conjecture about central elements in nilpotent centralisers. It would extend the known strange-orbit result discussed immediately before it.