  Given a complex manifold endowed with a $\mathbb{C}^\times$-action and a
DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action),
we prove that if the $\mathbb{C}^\times$-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.
