  We investigate a certain class of Ind-scheme morphisms corresponding to
homomorphisms between the automorphism group of the $n$-th complex Weyl algebra
and the group of Poisson structure-preserving automorphisms of the commutative
complex polynomial algebra in $2n$ variables. A conjecture of Kanel-Belov and
Kontsevich, whose proof we have recently obtained, states that these
automorphism groups are canonically isomorphic in characteristic zero, with the
mapping discussed here being the candidate for the isomorphism. The main
objective of the present paper is to establish the independence of the said
mapping of the choice of infinite prime - that is, the class $[p]$ of prime
number sequences modulo fixed non-principal ultrafilter $\mathcal{U}$ on the
index set of positive integers. To that end, we introduce the augmented and
skew augmented versions of algebras in question and study the augmented
Ind-morphism between the normalized automorphism Ind-schemes in the context of
tame automorphism approximation. In order to correctly implement approximation
in our proof, we study singularities of curves in skew augmented automorphism
Ind-schemes and their images under Ind-scheme morphisms. Apart from that, we
study the augmented version of the independence conjecture.
