Braided equivalence of module categories Rep(V_L^+) and Rep(V_K^+) in the A_{2n} lattice case

Establish a braided tensor equivalence between the modular tensor categories Rep(V_L^+) and Rep(V_K^+), where L is the root lattice of type A_{2n}, K is a positive definite even lattice chosen so that (L,K) embeds into an even unimodular lattice E with E decomposing as ⋃_{r=0}^{n}(L+λ_r, K+μ_r), and Rep(V_L) is braided equivalent to Rep(V_K). Concretely, prove that the fixed-point vertex operator algebras V_L^+ and V_K^+ have braided equivalent module categories.

Background

In the A_{2n} lattice setting, the paper recalls that there exists an even lattice K such that the lattice VOAs V_L and V_K have braided equivalent module categories. The authors then analyze fixed-point VOAs V_L^+ and V_K^+ under the -1 isometry and develop fusion action structures relating to orbifolds.

They conjecture that the braided equivalence persists after taking fixed points, i.e., between Rep(V_L^+) and Rep(V_K^+), but explicitly state that they cannot prove this equivalence within the paper.