Uniqueness of minimal reducing subalgebras for nilpotent elements

Let $\mathfrak g$ be a semisimple Lie algebra, let $f\in\mathfrak g$ be a non-zero nilpotent element, and let $\mathfrak s$ be an $\mathfrak{sl}_2$-triple containing $f$. A reducing subalgebra for $f$ in $\mathfrak g$ is a semisimple subalgebra $\mathfrak q$ of $\mathfrak g$, normalized by $\mathfrak s$, such that $Z(\mathfrak s)(\mathfrak q\cap\mathfrak g_{\tilde d})$ contains a non-empty Zariski open subset, where $\tilde d$ is the reduced depth of $f$ and $Z(\mathfrak s)$ is the centralizer of $\mathfrak s$ in the corresponding semisimple algebraic group. Uniqueness conjecture for minimal reducing subalgebras. For any non-zero nilpotent element $f$ and $\mathfrak{sl}_2$-triple $\mathfrak s$ containing it, there is a unique minimal reducing subalgebra, up to conjugacy by $Z(\mathfrak s)$. The conjecture concerns the canonical structure associated with cyclic elements of nilpotent type and remains unresolved in the supplied source context.