Geometric Langlands duality for the spectral period of the determinantal cone

Let $C$ be the underlying curve and let $G=\operatorname{GL}_2\times\operatorname{GL}_2$, with Langlands dual group $\check{G}$. Define

$$\operatorname{Loc}^{\check{X}}_{\check{G}}=\operatorname{Map}(C_{\operatorname{dR}},\check{X}/\check{G}),$$

whose $R$-points are triples $(F_1,F_2,s)$, where $F_1,F_2\in\operatorname{Loc}_2(R)$ are rank-$2$ vector bundles on $C_R$ with flat connections along $C$, and $s$ is a flat section of $F_1\otimes F_2$ landing pointwise in the locus of tensors of rank at most $1$. Let

$$\mathcal{L}_{\check{X}}\cong\left(\pi^{\mathrm{spec}}_*\omega_{\operatorname{Loc}^{\check{X}}_{\check{G}}}\right)^{\mathbin{\mkern-6mu\fatslash}}\in\operatorname{IndCoh}_{\operatorname{Nilp}}(\operatorname{Loc}_{\check{G}})$$

be the co-localized spectral period sheaf. The proposed extension of Geometric Langlands duality. Under the Geometric Langlands equivalence for $G=\operatorname{GL}_2\times\operatorname{GL}_2$, one has

$$\operatorname{Dmod}(\operatorname{Bun}_G)\ni\mathcal{P}_X\,\langle 8(g-1)\rangle\xmapsto{\mathbb{L}_G}\mathcal{L}_{\check{X}}\in\operatorname{IndCoh}_{\operatorname{Nilp}}(\operatorname{Loc}_{\check{G}}).$$

The claim extends the corresponding statement of Ben-Zvi, Sakellaridis, and Venkatesh to the singular space $\check{X}$, where their cited conjecture does not formally apply.