Moore–Tachikawa conjecture for the TQFT associated to a semisimple group

Let $G$ be a connected complex semisimple affine algebraic group with Lie algebra $\mathfrak{g}$. Let $(e,h,f)\in\mathfrak{g}^{\times 3}$ be a principal $\mathfrak{sl}_2$-triple, and let

$$\mathcal{S}\coloneqq e+\mathfrak{g}_f$$

be the associated Slodowy slice. Let $\mathrm{COB}_2$ be the symmetric monoidal category of two-dimensional cobordisms, and let $\mathrm{MT}$ be the symmetric monoidal category whose objects are complex semisimple affine algebraic groups and whose morphisms are the corresponding Hamiltonian varieties described above. Moore–Tachikawa conjecture. There exists a $2$-dimensional TQFT

$$\eta_G:\mathrm{COB}_2\longrightarrow\mathrm{MT}$$

satisfying $\eta_G(S^1)=G$ and sending the cup cobordism to $[G\times\mathcal{S}]$, where $G\times\mathcal{S}$ is equipped with the Hamiltonian $G$-variety structure specified in the source. The conjecture proposes a geometric representation of two-dimensional cobordisms through Hamiltonian reduction in the category $\mathrm{MT}$; the supplied source does not state a resolution, so its status remains open.