The E3-structure conjecture for the bubbling spherical Hecke category

Let $G$ be the reductive group underlying the spherical Hecke category, and let

$$\mathcal{H}_\mathit{sph}^\mathit{bub}=\mathcal{A}(\mathbb{P}^1)=\mathit{Sh}(\operatorname{Bun}_{G}(\mathbb{P}^1))$$

be the bubbling spherical Hecke category, equipped with its monoidal structure. The E3-structure conjecture. Bubbling at any point of $\mathbb{P}^1$ equips $\mathcal{H}_\mathit{sph}^\mathit{bub}$ with the structure of an $E_3$-category, and there is an equivalence of $E_3$-categories

$$\mathcal{H}_\mathit{sph}^\mathit{bub}\simeq\mathcal{H}_\mathit{sph}.$$

The conjecture refines the constructed monoidal equivalence by incorporating the fusion product and the expected higher structure on the spherical Hecke category.