Integral period relations conjecture for Hilbert base change

Let $F$ be the totally real field of degree $d$, let $f$ be a normalized newform and $f_F$ its normalized holomorphic base change. For $\epsilon\in\widehat{W_F}$, let $d^+$ and $d^-$ be the cardinalities of the sets of embeddings on which $\epsilon$ has values $+1$ and $-1$, respectively, so that $d^++d^-=d$. Let $\Omega_{f_F}^\epsilon$ be the $\epsilon$-period of $f_F$, and let $\Omega_f^+$ and $\Omega_f^-$ be the periods of $f$. Integral period relations conjecture. If $p$ is prime to the discriminant of $F$ and to $6N\varphi_F(N)$, then

$$\Omega_{f_F}^\epsilon\sim (\Omega_f^+)^{d^+}(\Omega_f^-)^{d^-}.$$

This predicts that the integral periods of the base-change form factor into powers of the two periods of the original modular form. The source does not state a resolution, although it later uses the relation to motivate further conjectures.