Does the constructed equivalence extend to G-crossed fusion categories?

Determine whether the natural equivalence constructed in Lemma gcrossedpointed between the pointed G-crossed braided fusion category P with trivial component B_z and the G-crossed braided zesting (Vec_G ⊠ B_z)^{(α, ν)} extends from an equivalence of underlying G-graded fusion categories to an equivalence of full G-crossed fusion categories, in the sense of respecting both the G-action functors and the G-braiding (i.e., the crossed structure). Concretely, ascertain whether the functor Φ(g, φ) = Z_g φ defines a braided equivalence of G-crossed fusion categories between P and (Vec_G ⊠ B_z)^{(α, ν)}.

Background

In the analysis preceding Lemma gcrossedpointed, the authors extract a 2-cocycle α and a family of isomorphisms ν from a pointed G-crossed braided fusion category P with P_e = B_z, thereby producing a G-crossed braided zesting (Vec_G ⊠ B_z)^{(α, ν)}. They then construct a functor Φ(g, φ) = Z_g φ and verify that it yields an equivalence at the level of G-graded fusion categories.

However, it is unclear whether this equivalence lifts to the full G-crossed structure, which includes compatibility with the G-action functors and the G-braiding (crossed braiding). The authors explicitly note their uncertainty and suspicion that such an equivalence may fail at the G-crossed level, framing a concrete unresolved question about the categorical structure preservation under this identification.