The conjectural reciprocity law for rigid meromorphic cocycles

Fix open subgroups $U^p\subseteq G(\mathbb{A}^{p,\infty})$ and $U_p\subseteq G(\mathbb{Q}_p)$ containing $G(\mathbb{Q}_p)^+$, and put $U=U^pU_pG(\mathbb{R})^+$. For a coset $h\in G(\mathbb{A})/U$, let $\Gamma_h=G(\mathbb{Q})\cap hUh^{-1}$. Let $(\vec{x},h)$ be an oriented adelic special point, let $C_{x,h}$ be its associated relative class group, let $H_{x,h}/E_x$ be the corresponding abelian extension, and let $E_U$ be the field appearing in the source. Write $t\star(\vec{x},h)$ for the class-group action and $\operatorname{rec}:C_{x,h}\to\operatorname{Gal}(H_{x,h}/E_x)$ for the Artin reciprocity map. Conjectural reciprocity law. For every rigid meromorphic cocycle $J\in\mathcal{RMC}(\Gamma_h)$ of level $\Gamma_h$, there exists a finitely generated submodule $\Pi_J\subseteq \mathbb{Q}_p^\times$ of rank less than or equal to the rank of $H^s(\Gamma_h,\mathbb{Z})$ such that

$$J[(\vec{x},h)]\in H_{x,h}E_U\Pi_J$$

for every oriented special pair $(\vec{x},h)$ at which $J$ is regular. Moreover, for every $\alpha\in\Pi_J$ such that $J[(\vec{x},h)]/\alpha\in H_{x,h}E_U$, there exists $\alpha'\in\Pi_J$ such that

$$\operatorname{rec}(t)\left(\frac{J[(\vec{x},h)]}{\alpha}\right)=\frac{J[t\star(\vec{x},h)]}{\alpha'}\quad\forall t\in C_{x,h}^{U}.$$

This is an analogue of Shimura reciprocity: the values of rigid meromorphic cocycles at adelic special pairs should be defined over the predicted class-field-theoretic extensions, up to the finitely generated ambiguity $\Pi_J$, with Galois action compatible with the class-group action. The source presents this as conjectural and gives no resolution.