$C_n$-cofinite twisted modules for $C_2$-cofinite vertex operator algebras

Given a vertex operator algebra $V$ with a general automorphism $g$ of $V$, we introduce a notion of $C_n$-cofiniteness for weak $g$-twisted $V$-modules. When $V$ is $C_2$-cofinite and of CFT type, we show that all finitely-generated weak $g$-twisted $V$-modules are $C_n$-cofinite for all $n\in {\mathbb{Z}}_{>0}$.