Real-analytic dependence on the disphenoid parameter

Establish whether the family of Smyth free-boundary minimal disks, denoted by $\Sigma^{*}_{\beta}$, depends real-analytically on the disphenoid parameter $\beta$ in the relevant one-parameter family, particularly for $\beta\in(\tfrac12,\tfrac23)$.

Background

The paper treats the endpoint $\beta=2/3$, where the disphenoid is a Coxeter cell and the associated disks can be identified explicitly with classical triply periodic minimal surfaces. It does not resolve the behavior of the corresponding family away from that endpoint.

The unresolved issue is whether the parameterized family of free-boundary minimal disks has real-analytic dependence on $\beta$; the paper separately notes that claims about the interior members of the family remain unresolved.