Belov-Kanel–Kontsevich conjecture on Weyl algebra automorphisms

Let $n$ be a positive integer, let $A_n(\mathbb{C})$ denote the Weyl algebra of index $n$ over $\mathbb{C}$, and equip $\mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_{2n}]$ with its standard symplectic Poisson structure. Belov-Kanel–Kontsevich conjecture. The automorphism group of $A_n(\mathbb{C})$ is isomorphic to the group of Poisson automorphisms of $\mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_{2n}]$. Its importance is connected to other open problems, including the Jacobian conjecture and the Dixmier conjecture; the statement remains open.