Beilinson–Bloch conjecture for projected modified diagonal cycles

Let $F$ be a number field and let $X/F$ be a curve. Let $A_1,A_2,A_3$ be three simple isogeny factors of $\operatorname{Jac}_X$. Assume that the $L$-function

$$L\left(s,h^1(A_1)\otimes h^1(A_2)\otimes h^1(A_3)\right)$$

has holomorphic continuation to $s\in{\mathbb C}$. Beilinson–Bloch-type conjecture. If

$$L\left(2,h^1(A_1)\otimes h^1(A_2)\otimes h^1(A_3)\right)\neq 0,$$

then

$$[\Delta]_{h^{A_1}(X)\otimes h^{A_2}(X)\otimes h^{A_3}(X)}=0.$$

The source describes this as a special case of the Beilinson–Bloch conjecture; it concerns the vanishing of a motivically projected modified diagonal under analytic continuation and nonvanishing at the critical value, and is not resolved in the stated generality.