Affine torsor property for the full automorphism group of BK

Establish whether the quotient morphism from the stack of injections Inj(K,G) to Inj(K,G)/Aut(BK) is an affine Aut(BK)-torsor over the base stack M for every finite abelian group K and oriented P-divisible group G over M.

Background

The paper proves that the restricted Aut(K)-action on the stack of injections Inj(K,G) produces an affine flat Aut(K)-torsor over the stack of subgroups Sub(K,G). The larger group Aut(BK), however, acts naturally before truncation, and this action is geometrically more canonical.

It is unknown in general whether the quotient by Aut(BK) is affine. The authors note that examples involving multiplicative groups and elliptic curves suggest that the full action may fail to define an affine torsor, but no explicit counterexample is known because the available examples require inverting the relevant integer, making the associated Galois extensions comparatively uninformative.