The braided equivalence conjecture for symplectic fermion representation categories

Let $\beta=e^{-\pi i d/8}$, let $\mathcal{S}\hspace{-.65pt}\mathcal{F}(\mathfrak{h})$ be the symplectic fermion category, and let $\mathbb{V}(d)_\mathrm{ev}$ be the even subalgebra of the corresponding vertex operator algebra. Define the functor

$$F:\mathcal{S}\hspace{-.65pt}\mathcal{F}(\mathfrak{h})\longrightarrow\operatorname{Rep}(\mathbb{V}(d)_\mathrm{ev}),\qquad X\longmapsto(\widehat X)_\mathrm{ev}.$$

Braided equivalence conjecture. For $\beta=e^{-\pi i d/8}$, the functor $F$ is a $\mathbb{C}$-linear braided monoidal equivalence. This conjecture identifies the representation category of the even symplectic fermion vertex operator algebra with the braided tensor category constructed from symplectic fermions.