B-orbits in abelian nilradicals of types B, C and D: towards a conjecture of Panyushev

Let $B$ be a Borel subgroup of a semisimple algebraic group $G$ and let $\mathfrak{m}$ be an abelian nilradical in $\mathfrak{b}=\mathrm{L}\mathrm{i}\mathrm{e}(B)$. Using subsets of strongly orthogonal roots in the subset of positive roots corresponding to $\mathfrak{m}$, D. Panyushev \cite{Pan} gives in particular classification of $B-$orbits in $\mathfrak{m}$ and ${\mathfrak{m}}^{\ast }$ and states general conjectures on the closure and dimensions of the $B-$orbits in both $\mathfrak{m}$ and ${\mathfrak{m}}^{\ast }$ in terms of involutions of the Weyl group. Using Pyasetskii correspondence between $B-$orbits in $\mathfrak{m}$ and ${\mathfrak{m}}^{\ast }$ he shows the equivalence of these two conjectures. In this Note we prove his conjecture in types $B_n,C_n$ and $D_n$ for adjoint case.