Strange-orbit conjecture for distinguished partitions in ${\mathfrak{sl}}_n$

For every $n\geqslant3$, let ${\mathcal O}(\boldsymbol{\lambda})\subset{\mathfrak{sl}}_n$ be the unique nilpotent orbit with

$$\dim{\mathcal O}(\boldsymbol{\lambda})=\dim{\mathfrak b}+{\rm ind}\,{\mathfrak b},\qquad \lambda_1=3,\qquad \lambda_2<3,$$

where ${\mathfrak b}$ is a Borel subalgebra and $\boldsymbol{\lambda}$ is its partition. Strange-orbit conjecture. The orbit ${\mathcal O}(\boldsymbol{\lambda})$ is strange, and a complementary subalgebra is a Frobenius parabolic subalgebra of minimal dimension.

The source gives the partitions explicitly for even and odd $n$ and notes that these orbits are not spherical because $\lambda_1>2$. The conjecture therefore concerns a family of non-spherical strange nilpotent orbits and their complementary subalgebras.