2 problems

Invariant-abelian-subalgebra conjecture. The parameters satisfy $\lambda=\mu=0$. Equivalently, $\mathfrak{g}$ decomposes into three orthogonal $\Psi$-invariant abelian subalgebras.

S^3-action rigidity conjecture. If $\Psi$ has no nonzero singular eigenvectors, then $\Psi$ is a known example coming from an $S^3$-action.