Lattice-realization conjecture for even cyclic permutations of free fermions

Let $V_{fer}$ denote the free fermion vertex operator superalgebra, let $k$ be an even positive integer, and let $V_{\mathbb{Z}\alpha}$ be the rank-one lattice vertex operator superalgebra corresponding under boson–fermion correspondence to two free fermions. The lattice-realization conjecture. The $(1\;2\;\cdots\;k)$ permutation automorphism of $V_{fer}^{\otimes k}$ is conjugate, under boson–fermion correspondence, to a lift of a lattice isometry on $V_{\mathbb{Z}\alpha}^{\otimes k/2}$. The conjecture would identify the permutation-twisted free-fermion construction with a lattice-isometry-twisted construction; the paper establishes the two-fermion case, while the general even case remains open.