The modular tensor category conjecture for even symplectic fermions

Let $\mathcal{V}_{\mathcal{SF}_n^{\mathrm{even}}}$ be the even part of the vertex superalgebra of $n$ pairs of symplectic fermions, and let $\mathcal{C}_0\oplus\mathcal{C}_1$ be the category used in the source, with odd anticommuting generators $E_i,F_i$. The modular tensor category conjecture for even symplectic fermions. The category $\mathcal{V}_{\mathcal{SF}_n^{\mathrm{even}}}\text{-}\operatorname{mod}$ is equivalent, as a modular tensor category, to the abstract braided monoidal category $\mathcal{SF}_n$ defined by Runkel on $\mathcal{C}_0\oplus\mathcal{C}_1$. As an abelian category,

$$\mathcal{V}_{\mathcal{SF}_n^{\mathrm{even}}}\text{-}\operatorname{mod}=\mathcal{C}_0+\mathcal{C}_1\cong \mathbb{C}[E_1,\ldots,E_n,F_1,\ldots,F_n](\operatorname{SVect})\text{-}\operatorname{mod}\oplus\operatorname{SVect},$$

where the polynomial algebra is generated by odd, pairwise anticommuting elements $E_i,F_i$; under this equivalence, $\chi_3,\chi_4$ correspond to $\mathbb{C}^{1|0},\mathbb{C}^{0|1}$ in $\mathcal{C}_1=\operatorname{SVect}$, while $\chi_1,\chi_2$ correspond to $\mathbb{C}^{1|0},\mathbb{C}^{0|1}$ in $\mathcal{C}_0$ with all $E_i,F_i$ acting by zero. This conjecture proposes an explicit description of the representation theory of even symplectic fermions, including its braided tensor structure; the source states no proof or resolution, although it notes that the abelian-category part is the more accessible aspect.