Study on finiteness properties of vertex operator algebras

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I have been working on a finiteness property called a C_2-cofiniteness property, for a vertex operator algebras. A fixed point subalgebra of a vertex operator algebra by its finite automorphism group is called an orbifold model. In the representation theory of vertex operator algebra there is a conjecture that if a vertex operator algebra has the C_2-cofiniteness property then so does any orbifold model of it. I have found necessary and sufficient conditions which make the conjecture true for commutative vertex algebras and Z_2-permutation orbifold models of vertex operator algebras. I also showed that Z_2-permutation orbifold models of the Virasoro vertex operator algebras having C_2-cofinittness property satisfy the C_2-cofiniteness condition.