The first $\gamma,\delta$ conjecture for endomorphisms of the Weyl algebra

Let $A_1$ be the first Weyl algebra, let $f$ be an endomorphism of $A_1$, and let an involution mean an automorphism or anti-automorphism of order two. Two involutions $\gamma$ and $\delta$ make $f$ a $\gamma,\delta$-endomorphism when

$$f\circgamma=\delta\circ f.$$

Here $\alpha$ is the involution fixed earlier in the paper, and an involution is conjugate to $\alpha$ when it is obtained from $\alpha$ by conjugation by an automorphism or anti-automorphism of $A_1$.

The first $\gamma,\delta$ conjecture. For every endomorphism $f$ of $A_1$, there exist involutions $\gamma$ and $\delta$, each of $\gamma$ and $\delta$ is conjugate to $\alpha$, such that $f$ is a $\gamma,\delta$-endomorphism of $A_1$.

The paper proves that this conjecture is equivalent to the Dixmier conjecture for $A_1$, so it is an alternative formulation of the assertion that every endomorphism of $A_1$ is an automorphism. Its resolution is therefore not established by the supplied text.