Factorization conjecture for the $\mathbf{g}$-dominant triple product $p$-adic $L$-function

Suppose that $\varepsilon(\textup{\bf f})=-1$ and the stated hypothesis holds. Let ${\rm Log}_{\omega_\textup{\bf f}}({\rm BK}_\textup{\bf f}^\dagger)$ denote the logarithm of the big Beilinson--Kato class. Factorization conjecture. The $p$-adic $L$-functions satisfy

$$\mathcal{L}_p^\mathbf{g}(\textup{\bf f}\otimes\mathbf{g}\otimes\mathbf{g}^c)^2(\kappa,\lambda,\lambda)=\mathscr C(\kappa)\cdot\mathcal{L}_p^{\rm ad}(\textup{\bf f}\otimes {\rm ad}^0\mathbf{g})(\kappa,\lambda)\cdot {\rm Log}_{\omega_\textup{\bf f}}({\rm BK}_\textup{\bf f}^\dagger),$$

where $\mathscr C\in\mathcal R_\textup{\bf f}[1/p]$ is the specified factor. This is proposed as the $p$-adic family analogue of the complex derivative factorization, but the source gives no resolution of the conjecture.