Holomorphic Frobenius actions for DQ-modules

Given a complex manifold endowed with a Gm-action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the Gm-action is free and proper, then the category of F-equivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold.