Intuitively, I would say no. There are lots of other foldings. Take any twice differentiable function $f:[a,b]\mapsto\mathbb R$ where $a,b\in\mathbb R$, such that $f''(x)<0$ everywhere. Embed the $x$-$y$-plane with the graph $\Gamma$ of $f$ in $\mathbb R^3$ and take for each point of $\Gamma$ the plane perpendicular to $\Gamma$ (i.e. perpendicular to the tangent of $\Gamma$ in its original $x$-$y$-plane – think of $\Gamma$ as a sausage ). For any sphere with its center in the $x$-$y$-plane above $\Gamma$ and small enough not to intersect with $\Gamma$, these planes yield a folding. The condition $f''(x)<0$ ensures that the planes do not intersect each other above $\Gamma$, so the circles are disjoint.