Beilinson–Bloch conjecture for decomposable triple product diagonal cycles

Let $F$ be the totally real field, let $X_U$ be the Shimura curve, and let $\pi_1,\pi_2,\pi_3$ be the automorphic representations and $[\Delta_3]_{f_1\otimes f_2\otimes f_3}$ the corresponding modified diagonal-cycle class. Suppose $\pi_1\simeq\pi_2^\vee$, and let $\pi_3$ be self-dual with root number $\epsilon(1/2,\pi_3)=-1$. Let $(A,\iota)$ correspond to $\pi_3$. Beilinson–Bloch conjecture for decomposable triple product diagonal cycles. There exists a zero cycle $Z$ on $X_U$ such that

$$[\Delta_3]_{f_1\otimes f_2\otimes f_3}$$

is a multiple of

$$[\Delta_2]_{f_1\otimes f_2}\times {\mathfrak{pr}}_{\iota,*}f_{3,*}[Z].$$

Moreover, $Z$ can be chosen to be a CM $0$-cycle associated to some quadratic CM extension of $F$. This conjecture describes the expected relation between modified diagonal cycles and CM cycles in the decomposable case; the surrounding discussion motivates it using Gross–Zagier formulas and the predicted one-dimensionality of the relevant Chow group.