Motivic action conjecture for Hilbert modular varieties

Let $f$ be a parallel weight-one form, let $U_f^\vee$ be the dual space appearing in the cited construction, and for $u\in U_f^\vee$ let $u_i$ denote its $\sigma_i$-component. Let $\omega_f^{\{i\}}$ be the corresponding cohomological forms and let $\tau\otimes\iota$ be the pairing used in the source. Motivic action conjecture for Hilbert modular varieties. For every $u\in U_f^\vee$ outside the kernel of the cited pairing,

$$2\pi i\sum_{i=1}^d\frac{\omega_f^{\{i\}}}{\log(|\tau\otimes\iota(u)|)}\in H^1(X(\Gamma),\omega)$$

defines a cohomology class over $\overline{\mathbb{Q}}$. This is presented as a variant of Horawa's conjecture on rationality under partial complex conjugations, and the paper states that it has not been successfully approached.