Geometric origin of invariant categories under finite group actions

Determine whether, for any smooth proper S-linear category C of geometric origin and any finite group G acting S-linearly on C, the invariant category C^G is itself of geometric origin. In particular, resolve this question in the case C = D(X) for a smooth proper S-scheme X.

Background

The paper studies actions of finite groups on smooth proper S-linear categories and investigates when the corresponding invariant categories arise from geometry. Although C^G is always smooth and proper, its being of geometric origin (i.e., admitting a realization as a semiorthogonal component of D(Y) for a smooth proper S-scheme Y) is subtle.

The authors provide positive results by geometrizing many actions via gerbes and show that in these situations C^G is of geometric origin. However, they point out that the general case is not settled even for the fundamental geometric example C = D(X).