Kontsevich's Weyl–Poisson automorphism conjecture

Let $W_{n,\mathbb{C}}$ be the $n$-th Weyl algebra over $\mathbb{C}$, and let $P_{n,\mathbb{C}}$ be the $n$-th commutative Poisson algebra. Kontsevich's conjecture. The automorphism groups are isomorphic:

$$\operatorname{Aut}W_{n,\mathbb{C}}\simeq\operatorname{Aut}P_{n,\mathbb{C}}.$$

This conjecture is part of the proposed correspondence between Weyl algebra automorphisms and polynomial symplectomorphisms, motivated by deformation quantization. Its status is unresolved in the supplied text.