Moore–Tachikawa's TQFT conjecture in the algebraic moment-map category

Let $G$ be a connected complex 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 $\mathbf{AMT}$ be the algebraic moment-map category, and let $T^*G\to\mathfrak{g}^*$ denote the corresponding moment-map object. Moore–Tachikawa's conjecture in $\mathbf{AMT}$. There exists a TQFT $\eta_{T^*G}:\mathbf{Cob}_2\longrightarrow\mathbf{AMT}$ satisfying

$$\eta_{T^*G}(S^1)=(T^*G\longrightarrow\mathfrak{g}^*)$$

and assigning to the cup cobordism $G\times\mathcal{S}^{\vee}$. This is the algebraic moment-map reformulation obtained by composing the conjectural TQFT with the functor from $\mathbf{MT}$ to $\mathbf{AMT}$; the source gives no resolution of the conjecture.