Quaternionic pairing conjecture for logarithms of algebraic numbers

Let $F$ be a totally real number field of narrow class number $1$, let $\mathfrak p$ be a prime of $F$, and let $K_i/F$ be quadratic extensions in which $\mathfrak p$ is inert, with the stated almost totally complex signature conditions. Let $\mathcal{O}_i$ be orders in $K_i$, let $\psi_i\in\mathcal{E}(\mathcal{O}_i,R)$ be optimal embeddings, and let $J_{\psi_1,\psi_2}^{\bullet}$ be the evaluation pairings defined from $\varphi_{\psi_1}^0$, $c_{\psi_2}^0$, and $\Phi_{\psi_1}^{\bullet}$ for $\bullet\in\{\mathrm{even},\mathrm{odd},+,-\}$. Assume that the Hecke operators used to define $\varphi_{\psi_1}^0$ and $c_{\psi_2}^0$ are integers. Let $H_i$ be the narrow ring class field of $\mathcal{O}_i$ and set $H=H_1H_2$. Quaternionic pairing conjecture. There exist elements $P_{\psi_1,\psi_2}^{\mathrm{even}},P_{\psi_1,\psi_2}^{\mathrm{odd}},P_{\psi_1,\psi_2}^{+},P_{\psi_1,\psi_2}^{-}\in H$ such that

$$J_{\psi_1,\psi_2}^{\mathrm{even}}=\log_p(P_{\psi_1,\psi_2}^{\mathrm{even}}),\quad J_{\psi_1,\psi_2}^{\mathrm{odd}}=\log_p(P_{\psi_1,\psi_2}^{\mathrm{odd}}),\quad J_{\psi_1,\psi_2}^{+}=\log_p(P_{\psi_1,\psi_2}^{+}),\quad J_{\psi_1,\psi_2}^{-}=\log_p(P_{\psi_1,\psi_2}^{-}).$$

This is the paper's explicit formulation of the expectation that the evaluation pairings are logarithms of algebraic numbers. The source notes that the integer-Hecke hypothesis can always be achieved when $H_1(\Gamma_0,\mathbb{Z})$ is torsion, equivalently when the associated Shimura curve has genus $0$.