The classical Lie algebra conjecture on centralizers and sheets

Let $\mathfrak{g}$ be a real or complex classical semisimple Lie algebra. For $\mu\in\mathfrak{g}$, write $\mathfrak{g}^{\mu}$ for its centralizer, and let $S$ denote a sheet through $\mu$. Using the Killing-form orthogonal complement $V^{\bot}$ of a subspace $V\subseteq\mathfrak{g}$, the conjecture is:

Classical Lie algebra sheet conjecture. If there is only one sheet $S$ passing through $\mu$, then

$$[\mathfrak{g}^{\mu},\mathfrak{g}^{\mu}]=(\mathrm{T}_{\mu}S)^{\bot}.$$

If $\mu$ belongs to several sheets $S_{1},\dots,S_{k}$, then

$$[\mathfrak{g}^{\mu},\mathfrak{g}^{\mu}]=\left(\sum_{i=1}^{k}\mathrm{T}_{\mu}S_{i}\right)^{\bot}.$$

The paper proves the corresponding equality for compact Lie algebras and for $\mathfrak{sl}(n,\mathbb{C})$; the assertion for all classical semisimple Lie algebras is left as a conjecture.