The polyhedron-bisecting curve conjecture

Let $P$ be a polyhedron in $\mathbb{R}^3$, and let $\mathcal{C}$ be a smooth simple closed curve.

Polyhedron-bisecting curve conjecture. For each polyhedron $P$ in $\mathbb{R}^3$ there exists a smooth simple closed curve $\mathcal{C}$ such that

• $\mathcal{C}$ divides each face of $P$ into exactly two parts;
• $\mathcal{C}$ does not pass through the vertices of $P$;
• $\mathcal{C}$ is not tangent to the edges of $P$.

The source presents this as an open problem motivated by the greater apparent flexibility of cutting polyhedral faces in three dimensions than of cutting planar polygons.