Existence conjecture for adjoint triple-product $p$-adic $L$-functions

Let $y=(\kappa,\lambda)$ be an arithmetic, not necessarily crystalline, specialization, and let $?=\textup{\bf f},{\rm ad}$. The quantities $\mathcal E_p^?(\kappa,\lambda,\operatorname{ad})$ are the modified Euler factors defined in the surrounding text, and $C_{\textup{\bf f}_\kappa}^-$ is a $p$-adic period. Existence conjecture. There exist $p$-adic $L$-functions

$$\mathcal{L}_p^?(\textup{\bf f}\otimes {\rm ad}^0\mathbf{g})\in\mathcal R_2$$

with the stated piecewise interpolation formulas for $?=\textup{\bf f}$ and $?={\rm ad}$ at every such specialization $y$, including the Gauss sum and modified Euler factor specified in the source. The conjecture is motivated by Hsieh's $p$-optimal construction and the Coates--Perrin-Riou formalism; no resolution is supplied here.