The weak flatness conjecture for induced nilpotent orbits in orthogonal Lie algebras

Let $\fg$ be an orthogonal Lie algebra of type $B$ or $D$, of rank $n$, and write a Levi subalgebra as

$$\fl=\fg(a)\times\prod_{i=1}^{k}\fgl_{b_i},\qquad a+b_1+\cdots+b_k=n,$$

with $b_1\geqslant\cdots\geqslant b_k\geqslant1$. Let $\bO_e=\operatorname{Ind}_{\fl}^{\fg}(\{0\})$ be the induced orbit. The weak flatness conjecture. If $\fg$ is of type $B$, the weak flatness condition holds if and only if $a\geqslant b_1-1$. If $\fg$ is of type $D$, it holds if and only if at least one of $a\geqslant b_1-1$, $k=1$, or $b_1=2$ is satisfied. When $a=0$, the latter two conditions mean that the partition of $e$ has two rows or two columns. The conjecture is proved in the paper for $b_1=1,2$; the general case is obstructed by the conjectural description of generators of $\operatorname{gr} I_x$ when $a\neq0$.