Branch-point characterization of focusing mKdV finite-gap solutions

Consider the focusing mKdV hierarchy in the cases

$$\mathfrak{g}=\mathfrak{su}(2)\quad\text{or}\quad\mathfrak{g}=\imath\mathfrak{su}(2),$$

with the corresponding reality conditions

$$\alpha_{2m}\in\imath\operatorname{\mathbb{R}},\qquad \beta_{2m-1}=-\bar{\gamma}_{2m-1},\qquad \mathrm{b}\in\imath\operatorname{\mathbb{R}},$$

or

$$\alpha_{2m}\in\imath\operatorname{\mathbb{R}},\qquad \beta_{2m-1}=\bar{\gamma}_{2m-1},\qquad \mathrm{b}\in\operatorname{\mathbb{R}},$$

for $m=0,\ldots,N-1$. Let $\mathcal{V}$ be the associated hyperelliptic curve, and let $\mathfrak{J}^{\operatorname{\mathrm{Im}}}_K$ and $\mathfrak{J}^{\operatorname{\mathrm{Re}}}_K$ denote the relevant imaginary and real components of its Jacobian.

Branch-point characterization. All expressions given by the corresponding finite-gap formula comply with the first reality condition on $u\in\mathfrak{J}^{\operatorname{\mathrm{Im}}}_K$ and with the second reality condition on $u\in\mathfrak{J}^{\operatorname{\mathrm{Re}}}_K$ if and only if the branch points of $\mathcal{V}$ are $0$, $\infty$, and $g$ complex-conjugate pairs.

This characterizes the branch-point configuration producing the focusing mKdV reality conditions for the finite-gap expressions. The supplied text does not state whether this claim is proved or remains open.