Universal identities for Euclidean mechanical Lie brackets

Over $\mathbb{R}$, let $\mathfrak{P}$ be the class of Lie algebras $L = \bigoplus_{n \ge 0} L_n$ with $$[L_n, L_m] \subseteq L_{n+m-1} \quad (n + m > 0), \qquad [L_0, L_0] = 0,$$ and let $F$ be the free Lie algebra in this class on $A$ of degree two and $B$ of degree zero: every degree-preserving assignment of $A$ and $B$ extends uniquely to a homomorphism. For $d \ge 1$ and $V \in C^\infty(\mathbb{R}^d; \mathbb{R})$ let $\Phi_V : F \to C^\infty_{\mathrm{pol}}(\mathbb{R}^{2d})$ be the homomorphism with $$\Phi_V(A) = \tfrac12 |p|^2, \qquad \Phi_V(B) = V(q),$$ where the target consists of the smooth functions of $(q, p)$ that are polynomial in $p$, graded by degree in $p$, with the Poisson bracket $$\{f, g\} = \sum_{j=1}^d \bigl(\partial_{q_j} f\, \partial_{p_j} g - \partial_{p_j} f\, \partial_{q_j} g\bigr).$$ McLachlan and Murua, who call $F$ the Lie algebra of classical mechanics, showed that it is the direct sum of an abelian part $F_0$, isomorphic under $x * y = [y, [x, A]]$ to the free commutative nonassociative algebra on $B$, and the free Lie algebra on $A$ and the elements $[x, A]$, $x$ running over a basis of $F_0$. For a particular potential, $\Phi_V$ typically has a nonzero kernel. They showed that no nonzero element of $F_0$, and no nonzero element $[x, A]$ with $x \in F_0$, lies in the kernel of every $\Phi_V$.

Problem. Is $$\bigcap_{d \ge 1}\ \bigcap_{V \in C^\infty(\mathbb{R}^d; \mathbb{R})} \ker \Phi_V = \{0\}?$$ The intersection is over all dimensions and all potentials; injectivity of $\Phi_V$ for an individual potential is not asserted.