The injectivity conjecture for the classical-mechanics tree representation

Let $L_{\mathfrak P}(A,B)$ be the universal Lie algebra of classical mechanics generated by $A$ and $B$, and let $\mathcal T$ be the vector space spanned by the relevant classical-mechanics trees. Consider the linear map

$$\Theta:L_{\mathfrak P}(A,B)\longrightarrow \operatorname{span}(\mathcal T).$$

Tree-representation injectivity conjecture. The map $\Theta$ is injective.

This is presented as an equivalent formulation of the kernel-zero conjecture. The preceding proposition establishes the analogous statement for polynomial potentials, while injectivity in the full setting remains conjectural.