Classification conjecture for codimension-four \mathfrak{bms}_3 Poisson pairs

Let $\mathfrak{bms}_3^*$ be the dual of the centrally extended $\mathfrak{bms}_3$ algebra, let $c_1$ be its first central charge, and let $\{,\}_\mathrm{LP}^{\mathfrak{bms}_3^*}$ and $\{,\}_0^{\mathfrak{bms}_3^*}$ denote the Lie–Poisson and frozen Poisson brackets. For non-vanishing $c_1$, consider Poisson pairs of codimension $4$ on $\mathfrak{bms}_3^*$, classified by the type of their frozen point. Classification conjecture. All such Poisson pairs are classified into the following fifteen classes:

$$\begin{gathered} \left(\frac{dx^2}{2},0;\frac{dx^2}{2},0\right),\quad\left(dx^2,0;\frac{dx^2}{2},0\right),\quad\left(\frac{dx^2}{2},1;\frac{dx^2}{2},0\right),\quad\left(dx^2,1;\frac{dx^2}{2},0\right);\\ (0,1;0,1),\quad\left(\frac{dx^2}{2},1;0,1\right),\quad(0,2;0,1),\quad\left(\frac{dx^2}{2},2;0,1\right);\\ \left(\frac{dx^2}{2},1;\frac{dx^2}{2},1\right),\quad\left(dx^2,1;\frac{dx^2}{2},1\right),\quad\left(\frac{dx^2}{2},2;\frac{dx^2}{2},1\right),\quad\left(dx^2,2;\frac{dx^2}{2},1\right);\\ \left(\frac{dx^2}{2},0;0,0\right),\quad(0,1;0,0),\quad\left(\frac{dx^2}{2},1;0,0\right). \end{gathered}$$

The source presents this classification as an argument based on the relation between frozen points and codimension-four coadjoint orbits; it gives no resolution status beyond the stated claim.