3 problems

Panyushev's conjecture. Given orthogonal subsets $S,S'\subset\Psi(\mathfrak p^{\mathrm u})$, one has

Dimension-bound conjecture. There are no solvable compatible Lie extensions of $\mathcal{L}_s$ of dimension greater than $\underline{n_s}+5$.

Let $\mathfrak g$ be a simple Lie algebra, let $\mathfrak n$ be a nilradical in $\mathfrak g$, and let $U$ be the maximal unipotent subgroup associated with the chosen Borel subgro…