Factorization conjecture for geometric Langlands Hochschild cochains

Let $\mathbb C$ be the configuration-space curve, let $\operatorname{Ran}(\mathbb C)$ be its Ran space, and for $x\in\operatorname{Ran}(\mathbb C)$ let $\widetilde x\in\operatorname{Sym}^n(\mathbb C)$ range over lifts of $x$. For each lift, let $L_{\widetilde x}$ be the corresponding Levi subgroup and let $\operatorname{HC}^{\bullet}$ denote Hochschild cochains. Factorization conjecture. There exists a factorization algebra $\mathcal C$ over $\mathbb C$ whose stalk at $x\in\operatorname{Ran}(\mathbb C)$ is

$$\mathcal C_x=\bigoplus_{n\geq 1,\;\widetilde x\in\operatorname{Sym}^n(\mathbb C)}\operatorname{HC}^{\bullet}\bigl(\operatorname{IndCoh}_{\mathcal N_{L_{\widetilde x}}}(\operatorname{Flat}_{L_{\widetilde x}}(C))\bigr),$$

where $\widetilde x\in\operatorname{Sym}^n(\mathbb C)$ is a lift of $x\in\operatorname{Ran}(\mathbb C)$. This would encode how geometric Langlands categories for varying gauge groups assemble under collisions and configurations of D3 branes. The paper explains that the resulting operator-product structure would provide the relevant factorization data, but leaves its construction open.