An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron

Smyth (1984) proved that every tetrahedron $T\subset\mathbb{R}^3$ contains three embedded minimal disks meeting $\partial T$ orthogonally, obtained as conjugate surfaces of Plateau solutions in a skew quadrilateral $Q$ whose edges are $c_iν_i$, where $φ_1,φ_2,φ_3,φ_4$ are faces of $T$, $c_i=\operato