Incoherent Rankin–Selberg p-adic height nonvanishing conjecture

Assume Hypothesis $ref{hy:galois}$, let $\mathbb{V}_Pi$ be the associated Galois representation, and let $\mathscr{L}^1_\mathcal{E}(Pi)$ be the height-valued $p$-adic $L$-function. Let $\mathcal{E}/E$ be a $mathtt{P}$-extension containing $\mathcal{E}_{mathtt{P}'}$ for some nonempty $mathtt{P}'\subseteqmathtt{P}$. Height nonvanishing conjecture. If $\mathbb{V}_Pi\neq0$ and condition ($*$) holds, then $\mathscr{L}^1_\mathcal{E}(Pi)\neq0$. For $n=1$, the source says this follows from the cited work; the general assertion remains open.