  We conjecture that the category of permutation-twisted modules for a
multi-fold tensor product vertex operator superalgebra and a cyclic permutation
of even order is isomorphic to the category of parity-twisted modules for the
underlying vertex operator superalgebra. This conjecture is based on our
observations of the cyclic permutation-twisted modules for free fermions as we
discuss in this work, as well as previous work of the first author constructing
and classifying permutation-twisted modules for tensor product vertex operator
superalgebras and a permutation of odd order. In addition, we observe that the
transposition isomorphism for two free fermions corresponds to a lift of the -1
isometry of the integral lattice vertex operator superalgebra corresponding to
two free fermions under boson-fermion correspondence. We conjecture that all
even order cyclic permutation automorphisms of free fermions can be realized as
lifts of lattice isometries under boson-fermion correspondence. We discuss the
role of parity stability in the construction of these twisted modules and prove
that in general, parity-unstable weak twisted modules for a vertex operator
superalgebras come in pairs that form orthogonal invariant subspaces of
parity-stable weak twisted modules, clarifying their role in many other
settings.
