The lifting conjecture for endomorphisms of Weyl algebras

Let $A_n$ be the $n$'th Weyl algebra over ${\mathbb C}$, let $A_n^{(t)}$ be the quantized Weyl algebra over the parameter ring used in the paper, and let $\operatorname{End}(A_n)$ and $\operatorname{End}(A_n^{(t)})$ denote their algebra endomorphism monoids. A lift of $\phi\in\operatorname{End}(A_n)$ is an endomorphism $\widetilde{\phi}$ whose specialization at $t=1$ is $\phi$. Lifting conjecture. Any

$$\phi\in\operatorname{End}(A_n)$$

has a lift

$$\widetilde{\phi}\in\operatorname{End}(A_n^{(t)}).$$

The source gives evidence in dimension one: every automorphism of $A_1$ lifts, while the general assertion remains conjectural.