Congruence-ideal conjecture for non-base-change Hilbert cuspforms

Let $f$ be a normalized newform and $f_F$ its base change to $F$. Let $\eta_f^\sharp$ be the complementary factor in the factorization of the base-change congruence ideal, let $I_G$ denote the relevant character associated with the extension, and let $\operatorname{Irr}_F$ be the irreducibility hypothesis. Assume that $p$ does not divide $6dDN\phi_F(N)$, $(\operatorname{Irr}_F)$ holds, and the Hecke algebras $\mathcal{T}$ and $\mathcal{T}_F$ are Gorenstein. Congruence-ideal conjecture. One has

$$\eta_f^\sharp\sim L^\ast(\operatorname{Ad} f\otimes I_G).$$

In particular, if $p$ divides $L^\ast(\operatorname{Ad} f\otimes I_G)$, the source predicts a congruence between $f_F$ and a non-base-change Hilbert cuspform. The source says this will be proved in the real quadratic case; no general resolution is supplied.