The finite-subspace reality conjecture for trigonal sigma-functions

Let $\mathcal{V}$ be a trigonal curve whose finite branch points are real or occur in complex-conjugate pairs. Let $\wp_{i,j}$ and $\wp_{i,j,k}$ denote the associated Kleinian functions, let $u[\varepsilon]$ be the half-period determined by a half-integer characteristic $[\varepsilon]$, and let $g$ be the genus. The finite-subspace reality conjecture. There exist $2^g$ affine subspaces

$$\mathfrak{J}^{\operatorname{Re}}=\{\Omega+s\mid s\in\mathbb{R}^g\},$$

parallel to the real axes, with $\Omega=u[\varepsilon]$, such that $\wp_{i,j}(s+\Omega)$ and $\wp_{i,j,k}(s+\Omega)$ are real-valued and have poles. With the choice of periods specified by the rhombic period-lattice conjecture, the corresponding $2^g$ half-periods are purely imaginary, $\Omega\in\imath\mathbb{R}^g$. On

$$\mathfrak{J}^{\operatorname{Im}}=\{\imath s\mid s\in\mathbb{R}^g\},$$

the functions $\wp_{i,j}(\imath s)$ are real-valued, while $\wp_{i,j,k}(\imath s)$ are purely imaginary. These reality assertions describe the real and imaginary finite-gap subspaces on which the Kleinian functions provide real-valued data; their resolution is not specified in the source.