Canonical Planck-parameter lift conjecture for polynomial symplectomorphisms

Let $R$ be a finitely generated smooth commutative algebra over $\mathbb Z$, let $P_{n,R}$ be the polynomial Poisson algebra, and let $A^{\hbar}_{n,R[\hbar]}$ be the algebra over $R[\hbar]$ generated by $\hat{x}_1,\ldots,\hat{x}_{2n}$ with relations

$$[\hat{x}_i,\hat{x}_j]=\hbar\omega_{ij}.$$

For a positive integer $M$, write $R(M^{-1})$ for the localization obtained by inverting $M$. Planck-parameter lift conjecture. For every symplectomorphism $g\in\operatorname{Aut}(P_{n,R})$, there exist a positive integer $M$ and an automorphism $\widetilde g\in\operatorname{Aut}(A^{\hbar}_{n,R(M^{-1})[\hbar]})$ over $R(M^{-1})[\hbar]$ such that $\widetilde g\pmod{\hbar}=g$, and, for all sufficiently large primes $p$, $\widetilde g\pmod p$ preserves the subalgebra $R/pR[y_1^p,\ldots,y_{2n}^p]$. This proposes a one-parameter deformation lifting classical symplectomorphisms to automorphisms of the Weyl algebra with Planck parameter; the supplied source gives no resolution.