Kontsevich–Kanel-Belov rational Weyl–Poisson correspondence conjecture

Let $W_n(\mathbb{Q})$ be the $n$-th Weyl algebra over $\mathbb{Q}$ and let $P_n(\mathbb{Q})$ be the polynomial Poisson algebra in $2n$ variables with its canonical Poisson structure. Kontsevich–Kanel-Belov correspondence conjecture.

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

The conjecture expresses the expected correspondence between quantum Weyl-algebra automorphisms and classical Poisson automorphisms; the supplied text gives no resolution.