Independence of twisted representations and Zhu's algebras under inner automorphisms

Let $V$ be a vertex operator algebra, let $g$ be an automorphism of $V$, and let $h$ be an inner automorphism of $V$. Consider lower bounded logarithmic $g$-twisted and $gh$-twisted modules, together with their twisted Zhu's algebras.

Inner-automorphism invariance conjecture. There is a one-to-one correspondence between the simple objects in the category of lower bounded logarithmic $g$-twisted modules and the simple objects in the category of lower bounded logarithmic $gh$-twisted modules, and

$$A_g(V)\simeq A_{gh}(V).$$

Thus both the simple twisted-module theory and the twisted Zhu's algebra should be independent of the inner automorphism used to modify $g$. The preceding result establishes this invariance in the affine vertex-operator-algebra setting considered in the paper; the assertion for a general vertex operator algebra and automorphism is left open.