The automorphic gluing equivalence for the bubbling degeneration

Let $\alpha$ be the functor

$$\alpha:\mathcal{A}(X_-,Q)\otimes_{\mathcal{H}^{\mathit{bub},\otimes R}}\mathcal{A}(X_+,Q)\longrightarrow\mathcal{A}(\mathfrak{Y}_{>0})$$

constructed from the bubbling family, where $\mathcal{H}^{\mathit{bub},\otimes R}$ is the relevant bubbling Hecke category over $R$. The automorphic gluing conjecture. The functor $\alpha$ of Theorem~ is an equivalence. This conjecture predicts that the automorphic category of the glued family is recovered as the tensor product of the two automorphic module categories over the bubbling Hecke category.