Rank-one Bloch–Kato conjecture for the triple product motive

Let $f_1,f_2,f_3$ be the triple of modular forms in the paper, let $\mathrm{V}(\underline{\mathbf{f}})$ be the associated $l$-adic representation over $E_{\underline{\lambda}}$, and let

$$\Theta_{\underline{\mathbf{f}}}\in \mathrm{H}^{1}(\mathbf{Q},\mathrm{V}(\underline{\mathbf{f}})(-1))$$

be the Abel–Jacobi image of the Gross–Kudla–Schoen diagonal cycle. Rank-one Bloch–Kato conjecture. If $\Theta_{\underline{\mathbf{f}}}$ is non-zero, then the Bloch–Kato Selmer group

$$\mathrm{H}^{1}_{f}(\mathbf{Q}_{l},\mathrm{V}(\underline{\mathbf{f}}))$$

has rank $1$ over $E_{\underline{\lambda}}$. This is presented as a conjectural rank-one consequence toward the Bloch–Kato conjecture for the triple product motive; the supplied text gives no resolution, so its status remains open.