Harris's period rationality conjecture for weight-one Hilbert modular forms

Let $R_f$ be the Stark regulator matrix, let $A=R_f^{-1}$, and let $\omega_f^{\sigma_i}$ denote the standard archimedean cohomology classes. For each $j$, form the linear combinations determined by the columns of $A$ and their exterior powers.

Harris period conjecture. These combinations give rational bases of the corresponding coherent cohomology groups; in particular, $\omega_f^{\Sigma_\infty}/\det R_f$ is rational.

The paper states that this conjecture is equivalent to Stark's conjecture for $\operatorname{Ad}^0\varrho_f$, so it is a refinement of Stark's special-value conjecture.