The Dixmier conjecture (Dixmier 1968) asserts that every algebra endomorphism of a Weyl algebra 
over a field of characteristic
0 is an automorphism. In the differential-operator
realization, an endomorphism chooses new polynomial differential operators obeying the same commutation
relations. The conjecture requires that these operators
generate the whole algebra.
The all-dimensional conjecture is equivalent to the all-dimensional Jacobian conjecture, although the reductions can change the dimension (Adjamagbo and van den Essen 2007). It is also closely related to the Poisson conjecture.
Long (2026) reported an explicit nonautomorphic endomorphism of 
in an appendix to a proposed rank-two Poisson counterexample.
This would disprove the conjecture for 
and, by adjoining unchanged generators, for 
. The construction was developed with AI assistance and
checked by its author, but independent external verification had not been reported
as of Sep. 7, 2026. It does not by itself settle the smaller ranks.
