Frobenius-parabolic conjecture for simple Lie algebras

Let ${\mathfrak g}$ be a simple Lie algebra. A Lie subalgebra is Frobenius when its index is zero; let ${\EuScript M}_{\sf Fr}$ denote the corresponding maximal dimension invariant, and let ${\mathfrak b}$ be a Borel subalgebra. Frobenius-parabolic conjecture. (i) The value ${\EuScript M}_{\sf Fr}$ is attained on Frobenius parabolic subalgebras: if ${\rm ind}\,{\mathfrak h}=0$, then there is a Frobenius parabolic subalgebra ${\mathfrak p}$ such that ${\mathfrak h}\subset{\mathfrak p}$. (ii) If ${\rm ind}\,{\mathfrak b}=0$, then

$$\dim{\mathfrak h}\leqslant\dim{\mathfrak b}$$

for every Frobenius subalgebra ${\mathfrak h}\subset{\mathfrak g}$, and only spherical orbits are strange.

Part (i) is stated to hold for ${\mathfrak{sl}}_n$ in the examples, while the source notes that small-rank cases relevant to (ii) can be checked. The general assertions remain conjectural in the supplied text.