Rank-zero Bloch–Kato conjecture for triple product motives

Let $f_1,f_2,f_3$ be the three modular forms, let $\mathrm{V}_1,\mathrm{V}_2,\mathrm{V}_3$ be their associated Galois representations, and write $\mathrm{V}(\underline{\mathbf{f}})=\mathrm{V}_1\otimes\mathrm{V}_2\otimes\mathrm{V}_3$. Suppose that the central critical value satisfies

$$L(f_{1}\otimes f_{2}\otimes f_{3},2)\ne 0.$$

Rank-zero Bloch–Kato conjecture. Then

$$\mathrm{H}^{1}_{f}(\mathbf{Q},\mathrm{V}(\underline{\mathbf{f}})(-1))=0.$$

This is the rank-zero case of the Bloch–Kato conjecture, which predicts a relationship between the order of vanishing of the triple product $L$-function at $s=2$ and the rank of its Bloch–Kato Selmer group. The statement is presented as a conjecture in the paper and is not resolved there.