Kontsevich–Kanel-Belov correspondence conjecture for Weyl and Poisson automorphisms

Let $W_n(\mathbb{Q})$ be the $n$-th Weyl algebra over $\mathbb{Q}$, and let $P_n(\mathbb{Q})$ be the polynomial algebra in $2n$ variables over $\mathbb{Q}$ equipped with its standard Poisson structure.

Kontsevich–Kanel-Belov correspondence conjecture. The automorphism groups are isomorphic:

$$\operatorname{Aut} W_n(\mathbb{Q})\simeq\operatorname{Aut} P_n(\mathbb{Q}).$$

The conjecture is part of the proposed correspondence between Weyl algebra automorphisms and Poisson automorphisms, arising from the relationship between quantization and polynomial symplectomorphisms. Its resolution status is not specified in the supplied text.