The Dixmier conjecture for the first Weyl algebra

Let $K$ be a field of characteristic zero. The Dixmier conjecture. Every endomorphism of the first Weyl algebra

$$A_1(K):=K\{t,x\}/\langle xt-tx-1\rangle$$

is an automorphism. Here $A_1(K)$ is generated by $t,x$ with $xt=tx+1$. This is the original Dixmier problem, formulated as a question in 1968 and commonly called a conjecture; its resolution is not established in the supplied text.