The algebraicity conjecture for special values of rigid meromorphic cocycles

Let $G$ be the orthogonal group and let $\Gamma$ be the arithmetic subgroup under consideration. Write $s$ for the relevant cohomological degree, let $\mathcal{RMC}(\Gamma)$ denote the group of rigid meromorphic cocycles, and let $\vec{x}$ be an oriented special point at which $J$ is regular. For a finitely generated submodule $\Pi_J\subseteq \mathbb{Q}_p^\times$, write $\overline{\mathbb{Q}}^\times\Pi_J$ for the subgroup generated by $\overline{\mathbb{Q}}^\times$ and $\Pi_J$. Algebraicity conjecture. For every $J\in\mathcal{RMC}(\Gamma)$, there exists a finitely generated submodule $\Pi_J\subseteq \mathbb{Q}_p^\times$ such that

$$J[\vec{x}]\in \overline{\mathbb{Q}}^\times\Pi_J$$

for every oriented special point $\vec{x}$ at which $J$ is regular. Moreover, the rank of $\Pi_J$ is less than or equal to the rank of $\operatorname{rk}_{\mathbb{Z}}H^s(\Gamma,\mathbb{Z})$. This is a crude algebraicity statement: special values need not themselves be algebraic, but should differ from algebraic numbers by elements of a finitely generated $p$-adic module. The source does not provide evidence resolving the conjecture.