Moore–Tachikawa's TQFT conjecture

Let $G$ be a connected semisimple affine algebraic group with Lie algebra $\mathfrak{g}$. Consider a principal $\mathfrak{sl}_2$-triple $(e,h,f)\in\mathfrak{g}^{\times 3}$, and set $\mathcal{S}=e+\mathfrak{g}_f$. Let $\mathbf{Cob}_2$ be the category of two-dimensional cobordisms and $\mathbf{MT}$ the category whose objects are complex semisimple groups and whose morphisms are holomorphic symplectic varieties with Hamiltonian group actions. Moore–Tachikawa's conjecture. There exists a two-dimensional TQFT $\eta_G:\mathbf{Cob}_2\longrightarrow\mathbf{MT}$ satisfying

$$\eta_G(S^1)=G$$

and assigning to the cup cobordism the holomorphic symplectic $G$-variety $G\times\mathcal{S}^{\vee}$, where $\mathcal{S}^{\vee}$ is the image of the Slodowy slice under the Killing-form identification $\mathfrak{g}\cong\mathfrak{g}^*$. This conjecture predicts a TQFT encoding the Higgs branches associated with six-dimensional superconformal quantum field theories; its status is presented in the source as ongoing, with no resolution supplied here.