Antisymmetric and separative class forcing witnessing failure of Powerset in symmetric extensions

Construct a tame class forcing notion that is both antisymmetric and separative and that, together with a suitable choice of automorphism group and normal filter of subgroups forming a class symmetric system, yields symmetric extensions in which the powerset axiom fails to be preserved.

Background

The authors exhibit a tame class forcing and a class symmetric system for which the powerset axiom is not preserved in the symmetric extension, demonstrating that tameness of the underlying class forcing does not guarantee preservation of Powerset by the symmetric system.

Their specific counterexample uses a class forcing that is antisymmetric but not separative, and they note it can be modified to be separative but then not antisymmetric. What remains unknown is whether one can produce such a counterexample using a class forcing that simultaneously satisfies both properties: antisymmetry and separativity.