Let $S_1,S_2,S_3$ be three simple closed curves on the 2-sphere $\mathbb{S}^2$. (With no smoothness or rectifiability assumption)

For each $i$, let $M_i$ denote the minimal surface bounded by $S_i$, as provided by Douglas/Rado. Note that $M_i$ is contained in the interior of $\mathbb{S}^2$ in $\mathbb{R}^3$.

Suppose that the intersections of the $S_i$ follow the same pattern as the intersections of the equator of the earth with two distinct Meridian circles; that is, $S_i \cap S_j$ is a pair of points for every $i \neq j$, and $S_i$ separates the two points of $S_{i+1} \cap S_{i+2}$ for every $i$, where addition is modulo 3.

Question: Must $M_1 \cap M_2 \cap M_3$ be a single point?