The p-adic Artin formalism factorization for triple product L-functions

Let $\mathbf{f}$ and $\mathbf{g}$ be the Hida families, let $\operatorname{As}(\mathbf{g})$ denote the Asai representation, let $\breve{\mathbf{g}}$ denote the associated representation appearing in the Artin formalism decomposition, and let $\lambda$ be the corresponding character. For each admissible choice of $\bullet$, consider the associated $p$-adic $L$-functions.

The p-adic Artin formalism conjecture. In full generalities, there should be a factorization

$$\mathscr{L}_p^{\bullet}(\mathbf{f} \otimes \operatorname{As}(\mathbf{g})) = \mathscr{L}_p^{\bullet}(\mathbf{f} \otimes\breve{\mathbf{g}})\cdot\mathscr{L}_p(\mathbf{f} \otimes \lambda).$$

This is the expected $p$-adic analogue of the factorization of complex Artin $L$-functions. The source presents it as a conjectural factorization in broad generality, with the relevant interpolation properties and periods still requiring establishment.