DC₂: the Dixmier conjecture for the second Weyl algebra

DC₂ is the rank-two case of the family, so DC₂ is a special case of the Dixmier conjecture in rank n. It survived July 2026 for the same reason DC₁ did: nothing false feeds into it.

It sits above the plane Jacobian conjecture. The Vaserstein–Kac pullback at $n=2$ turns a non-invertible Keller map of the plane into a non-surjective endomorphism of $A_2$, so DC₂ implies JC₂. A proof of DC₂ would settle the plane case. A counterexample to JC₂ would refute DC₂.^[1]

The route through the Poisson conjecture does not reach it. The chain JC₄ ⇒ PC₂ ⇒ DC₂ needs PC₂, and Long claims an explicit counterexample to PC₂ (unrefereed). If that claim holds, the implication PC₂ ⇒ DC₂ says nothing, and DC₂ stays open.^[2] See PCₙ.

1. ↑ A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, §1.
2. ↑ C. D. Long, An Explicit Counterexample to the Rank-Two Poisson Conjecture, arXiv:2608.23777 (22 July 2026).