Canonical period comparison under totally real quadratic base change

Let $F$ be a totally real number field, let $f$ be a Hilbert newform of parallel even weight over $F$, let $F'$ be a totally real quadratic extension of $F$, and let $f'$ be the base change of $f$ to $F'$. The canonical periods $4\Omega_f^{\mathrm{can}}4$ and $4\Omega_{f'}^{\mathrm{can}}4$ are defined using the Petersson inner product and congruence numbers. Canonical period comparison conjecture. One has

$$\Omega_{f'}^{\mathrm{can}}=(\Omega_f^{\mathrm{can}})^2$$

up to a $p$-adic unit. This comparison relates the canonical periods appearing in the anticyclotomic $p$-adic $L$-functions before and after totally real quadratic base change; the surrounding argument indicates that such a comparison is expected from the factorization into the base form and its quadratic twist.