The period-lattice conjecture for Hilbert modular abelian varieties

Let $f$ be a Hilbert newform of parallel weight $2$ over the totally real field $F$, with Hecke field $K_f$ of degree $g$. Let $s=s_i$ be a sign vector corresponding to the real embedding $\sigma_i\colon F\hookrightarrow\mathbb C$, and let $V_{f,s}\subset\mathbb C^g$ be the two-dimensional $K_f$-subspace generated by the period vectors associated with $s$ and the all-positive sign. A lattice $\Lambda\subset V_{f,s}$ is a discrete subgroup spanning $V_{f,s}$ over the relevant real vector space. Period-lattice conjecture. For every choice of lattice $\Lambda\subset V_{f,s}$, there is an isomorphism of complex abelian varieties

$$\mathbb C^g/\Lambda\sim A_f(\mathbb C),$$

where $A_f$ is the abelian variety predicted by the Eichler–Shimura conjecture, under the embedding $\sigma_i$ corresponding to $s_i$. This conjecture would identify the complex tori constructed from the Hilbert modular periods with the expected abelian variety, thereby justifying the moduli-point construction. The source gives no evidence of a resolution.