Slodowy slice intersecting a given orbit "minimally"?

Let $mathfrak{g}$ be a complex semisimple Lie algebra. Is it true that for any $Xinmathfrak{g}$, there exists an $mathfrak{sl}_2$-triple $(e,h,f)$ in $mathfrak{g}$ such that

1. We have $Xin e+Z_{mathfrak{g}}(f)$, and
2. $dim Z_{mathfrak{g}}(X)=dim Z_{mathfrak{g}}(e)$?

($Z_{mathfrak{g}}(-)$ always the centralizer in $mathfrak{g}$. One can deduce from the above conditions that the conjugacy orbit of $X$ must then meet the slice transversally, like in classical Kostant section situation.)

When $X$ is regular this is the well-known result about Kostant section. When $mathfrak{g}=mathfrak{gl}_n$ this is also true and can be deduced from the rational form of $X$. It might be that this question can be answered by a simple reference, but thanks a lot in advance in all cases!