Even cyclic permutation conjecture for twisted modules of vertex operator superalgebras

Let $V$) be a vertex operator superalgebra and let $k$ be an even positive integer. Consider weak, weak admissible, ordinary, parity-stable, and parity-unstable modules for the cyclic permutation $(1\;2\;\cdots\;k)$ acting on $V^{\otimes k}$, and parity-twisted modules for $V$. The even cyclic permutation conjecture. The category of weak parity-stable $(1\;2\;\cdots\;k)$-twisted $V^{\otimes k}$-modules is isomorphic to the category of weak parity-stable parity-twisted $V$-modules; the corresponding weak admissible and ordinary subcategories are likewise isomorphic, and all the corresponding subcategories of parity-unstable invariant subspaces coincide. This conjecture proposes the even-order analogue of the known odd-order correspondence between permutation-twisted and untwisted modules, with the free-fermion constructions providing evidence; the general categorical classification remains open.