ensor power of $A_1$, $A_n := A_1^{\otimes n} =
\underbrace{A_1 \otimes \cdots \otimes A_1}_{n \text{ times}}$, is called the [*$n$’th Weyl algebra*]{}. The algebra $A_n$ is a simple Noetherian domain of Gel’fand-Kirillov dimension $\GK(A_n)=2n$, it is canonically isomorphic to the algebra of polynomial differential operators $K \langle X_1, \ldots, X_n, \partial_1, \ldots, \partial_n \rangle$ (where $\partial_i = \tfrac{\partial}{\partial x_i}$) via $X_i \mapsto X_i$, $Y_i \mapsto \partial_i$ for $i=1,\ldots,n$.

In his seminal paper [@Dix], Dixmier (1968) found explicit generators for the group $G = \Aut_K(A_1)$ of $K$-automorphisms of the Weyl algebra $A_1$. Namely, the group $G$ is generated by the obvious automorphisms: $$(X, Y) \mapsto (X, Y + \l X^n), \quad (X, Y) \mapsto (X + \l Y^n, Y), \quad  (X, Y) \mapsto (\mu X, \mu^{-1} Y)$$ where $\l \in K$, $\mu \in K^*$ and $n \in \Nat_{+}:=\{1, 2, \ldots\}$.

In [@Dix], Dixmier posed six problems: The first problem of Dixmier (in the list) asks if [*every endomorphism of the Weyl algebra $A_1$ is an automorphism*]{}, i.e., given elements $P, Q$ of $A$ such that $[P, Q]=1$, do they generate the algebra $A_1$? A similar problem but for the $n$’th Weyl algebra is called the [*Dixmier Conjecture*]{}. Problems 3 and 6 have been solved by Joseph [@josclA1] (1975), Problem 5 and Problem 4 (in the case of homogeneous elements) have been solved by Bavula [@Bav-DixPr5] (2005).

The Dixmier Conjecture implies the [*Jacobian Conjecture*]{} (see [@BCW]) and the inverse implication is also true (see [@Tsuchimoto-2005] and [@Belov-Konts-2007]); a short proof is given in [@Bav-DCJC-2005]; see also [@Adjam-vdEs-2007]).

In [@Bav-RenQn], it is shown that for each $K$-endomorphism $\phi: A_n \to A_n$ its image is very large, i.e., the left $A_{2n}$-module ${}^{\phi}{A_n}^{\phi}$ is a holonomic $A_{2n}$-module (where for all $a, b \in A_n$ and $c \in {}^{\phi}{A_n}^{\phi}$, $a \cdot c \cdot b := \phi(a) c \phi(b)$). In particular, it has finite length with simple holonomic fa