Existence of the holomorphic E(7)-orbifold of the Barnes–Wall lattice VOA

Let $V_L$ be the lattice vertex operator algebra associated with the 128-dimensional Barnes–Wall lattice, and let $E(7)$ act on $V_L$. A holomorphic conformal field theory has central charge $c$ equal to its central charge as a VOA, and its gap is the smallest positive conformal weight of a primary state. Holomorphic-orbifold conjecture. The holomorphic $E(7)$-orbifold of the 128-dimensional Barnes–Wall lattice VOA exists. It is a holomorphic CFT of central charge $c=128$ with gap 4. Such an orbifold would provide a holomorphic CFT with a large spectral gap. The claim relies on the preceding evidence that the anomaly vanishes for $m=7$ and on the resulting character computation, but remains unproved.