The second $\gamma,\delta$ 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 be an automorphism or anti-automorphism of order two. Two involutions $\gamma$ and $\delta$ make $f$ a $\gamma,\delta$-endomorphism when

$$f\circ\gamma=\delta\circ f.$$

The second $\gamma,\delta$ conjecture. For every endomorphism $f$ of $A_1$, there exist involutions $\gamma$ and $\delta$, such that $f$ is a $\gamma,\delta$-endomorphism of $A_1$.

The source explains that this version removes the requirement that the involutions be conjugate to $\alpha$. It depends on the unresolved-looking claim that every involution of $A_1$ is conjugate to $\alpha$; the text describes a proposed Bass--Serre-theoretic argument whose first step has not yet been checked.