Base change equivalence for Hilbert newforms

Let $F/E$ be a Galois extension with group $G=\operatorname{Gal}(F/E)$, and let $f$ be a Hilbert newform of $G$-invariant level $\mathfrak{N}$, weight $k$, and character $\chi$. For each $\sigma\in G$, let ${}^\sigma\!\pi_f$ and ${}^\sigma\!f$ denote the conjugates of the automorphic representation and newform, and let $\rho_{f,\lambda}$ be the associated $\ell$-adic representation; let $\rho_{f,\lambda}^\sigma$ be its $\sigma$-conjugate. Let $\mathfrak{N}'$ be a level over $E$ depending on the relative discriminant $\mathfrak{D}_{F/E}$ and $\mathfrak{N}$. Then the following are equivalent: Base change conjecture. (a) ${}^\sigma\!\pi_f\simeq\pi_f$ for every $\sigma\in G$; (b) ${}^\sigma\!f=f$ for every $\sigma\in G$; (c) $\rho_{f,\lambda}^\sigma=\rho_{f,\lambda}$ for every prime $\lambda$ in $L_f$ and every $\sigma\in G$; (d) there exists a Hilbert newform $\hat f$ of level $U_1(\mathfrak{N}')$ and weight $k$ over $E$ such that

$$\rho_{\hat f,\lambda'}|_{\operatorname{Gal}(\overline{{\mathbf Q}}/F)}=\rho_{f,\lambda}$$

for all primes $\lambda'$ in $L_{\hat f}$ and $\lambda$ in $L_f$ above $\lambda'$; (e) there exists such a form $\hat f$ over $E$ satisfying

$$L(f,s)=\prod_{\eta\in\widehat G}L\bigl(\hat f\otimes(\eta\circ\operatorname{Art}_F),s\bigr),$$

where $\operatorname{Art}_F:F^\times\backslash{\mathbf A}_F^\times\to G_F^{\rm ab}$ is the global Artin reciprocity map. This gives equivalent automorphic, classical, Galois-representation, and $L$-function characterizations of base change. The source does not provide evidence resolving the claim.