The rhombic period-lattice conjecture for trigonal curves

Let $\mathcal{V}$ be a trigonal curve whose finite branch points $\{(e_i,d_i)\}$ are real or occur in complex-conjugate pairs. Let $\omega$ and $\eta$ be its period matrices, let $\varkappa$ be the associated matrix, and let $\omega'_j$ denote the relevant period vectors. The rhombic period-lattice conjecture. The period matrices $\omega$ and $\eta$ can be chosen purely imaginary, $\varkappa$ is real, and the period lattice is formed by rhombic sublattices, with

$$\operatorname{Im}\omega'_j \text{ spanned by } \frac{1}{2}\operatorname{Im}\omega_k,\qquad k\in\overline{1,g},\quad j\in\overline{1,g}.$$

This conjecture gives the period and reality structure underlying the real finite-gap solutions of the Boussinesq hierarchy; its resolution is not specified in the source.