Belov–Kontsevich conjecture on Weyl algebra automorphisms and polynomial Poisson automorphisms

Let $A_{n,\mathbb{C}}$ be the Weyl algebra and let $P_{n,\mathbb{C}}$ be the commutative polynomial algebra equipped with its Poisson bracket. For an infinite prime $[p]$, let

$$\varphi_{[p]}:\operatorname{Aut}(A_{n,\mathbb{C}})\longrightarrow\operatorname{Aut}(P_{n,\mathbb{C}})$$

be the group homomorphism obtained as the direct limit of the morphisms $\varphi_{[p],N}$ described above, sending a Weyl algebra automorphism $f$ to the associated symplectomorphism $f^c_{\Theta}$. Belov–Kontsevich conjecture. The homomorphism $\varphi_{[p]}$ is a group isomorphism. This conjecture asserts that automorphisms of the Weyl algebra are completely captured by symplectic automorphisms of the corresponding polynomial Poisson algebra; the supplied text gives no resolution status, so it remains open here.