The Euclidean mechanical systems kernel conjecture

For each dimension $d\geq 1$, let $V\in C^{\infty}(\mathbb{R}^d)$ be a smooth potential and let $\Phi_V$ be the homomorphism from the universal Lie algebra of classical mechanics to the Lie algebra generated by the Euclidean kinetic energy $T(p)=\frac{1}{2}p\cdot p$ and $V$ under the canonical Poisson bracket. The universal identities include antisymmetry, the Jacobi identity, and the vanishing of the Lie bracket of two elements of degree $0$.

Euclidean kernel conjecture. Consider all smooth potentials $V\in C^{\infty}(\mathbb{R}^d)$ with arbitrary $d$. Then

$$\bigcap_{d\geq 1}\bigcap_{V\in C^{\infty}(\mathbb{R}^d)}\ker(\Phi_V)=0.$$

This is the precise formulation of the claim that no relations beyond the inherited universal identities are shared by all smooth Euclidean mechanical systems. The paper presents it as unproved.