Unique representatives conjecture for coadjoint orbits of maximal unipotent subgroups

Let $\Phi$ be a root system with positive roots $\Phi^+$ and a fixed total lexicographic order $\succ$ on $\Phi^+$. For $\lambda\in\mathfrak{n}^*$ and $\gamma\in\Phi^+$, define

$$A_{\lambda,\gamma}:=\left(\lambda([e_\alpha,e_\beta])\right)_{\alpha\in\Phi^+,\alpha\succeq\gamma,\beta\in\Phi^+},$$ $$B_{\lambda,\gamma}:=\left(\lambda([e_\alpha,e_\beta])\right)_{\alpha\in\Phi^+,\alpha\succ\gamma,\beta\in\Phi^+}.$$

Let $S$ be the set of $\lambda\in\mathfrak{n}^*$ such that every $\gamma\in\operatorname{supp}(\lambda)$ satisfies the rank condition specified earlier in the paper.

Unique representatives conjecture. For every coadjoint orbit, there exists a unique linear form $\lambda\in S$ lying on that orbit.

The preceding argument establishes this representative property for the simple group of type $F_4$ by comparing the explicitly counted forms in $S$ with the known number of coadjoint orbits over finite fields. The conjecture proposes that the same construction works for other root systems, giving a canonical representative in $S$ for every coadjoint orbit.