Universal coequalizer conjecture for nearby cycles of automorphic categories

Let $C\rightsquigarrow C_0$ be a degeneration of a smooth complex projective curve to a nodal curve, with marked points $S\rightsquigarrow S_0$ disjoint from the nodes. Let $\psi:\mathit{Sh}_\mathcal{N}(\operatorname{Bun}_G(C,S))\to\mathit{Sh}_\mathcal{N}(\operatorname{Bun}_G(C_0,S_0))$ be the nearby-cycles functor, and let $\psi^L$ denote its left adjoint. At each node of $C_0$, the normalization construction gives two branches and hence a pair of affine Hecke actions on $\mathit{Sh}_\mathcal{N}(\operatorname{Bun}_G(C_0,S_0))$. Universal coequalizer conjecture. The functor $\psi^L$ is the universal functor co-equalizing the pair of affine Hecke actions at each node of $C_0$. This conjectural description would identify the image of nearby cycles and provide the desired automorphic Verlinde formula. No resolution is stated in the source.