Functoriality of automorphic $cL$-invariants under base change

Let $E/F$ be the finite extension and let $S_p(cpi)$ denote the relevant set of primes above $p$. For a prime $\tilde{\mathfrak{p}}\in S_p(\pi)$, let $e_{\tilde{\mathfrak{p}}}$ and $f_{\tilde{\mathfrak{p}}}$ be its ramification index and residue degree in $E$, respectively, and let $\mathfrak{q}$ be a prime of $E$ above $\tilde{\mathfrak{p}}$. Base-change functoriality conjecture. One has

$$e_{\tilde{\mathfrak{p}}}f_{\tilde{\mathfrak{p}}}\cdot \mathcal{L}^{\operatorname{cyc}}(\pi,\tilde{\mathfrak{p}})=\mathcal{L}^{\operatorname{cyc}}(\pi_E,\mathfrak{q}).$$

The assertion predicts the precise transformation of cyclotomic automorphic $\mathcal{L}$-invariants under base change; the surrounding results establish related invariance statements under additional hypotheses, while the status of this formulation is not resolved in the supplied text.