The first $g,h$ conjecture for endomorphisms of the Weyl algebra

Let $A_1$ be the first Weyl algebra and let $f$ be an endomorphism of $A_1$. An automorphism is an invertible algebra homomorphism, and an anti-automorphism is an invertible algebra antihomomorphism. Let $\alpha$ be the involution fixed earlier in the paper; an $\alpha$-endomorphism commutes with $\alpha$, while an $\alpha$-anti-endomorphism intertwines $\alpha$ with the relevant order-reversing structure.

The first $g,h$ conjecture. For every endomorphism $f$ of $A_1$, there exist $g$ and $h$, both automorphisms, both anti-automorphisms or one is an automorphism and the other is an anti-automorphism, such that $h f g^{-1}$ is an $\alpha$-endomorphism (when both $g$ and $h$ are automorphisms or both are anti-automorphisms) or an $\alpha$-anti-endomorphism (when one of $g$ and $h$ is an automorphism and the other is an anti-automorphism).

The surrounding discussion presents this as another sufficient condition for the Dixmier conjecture, and the preceding argument identifies the corresponding formulation with automorphy of every endomorphism. The supplied text does not report a resolution.