A classical theorem of Whitney states that the 1-skeleton of every triangulation of the 2-sphere $\mathbb{S}^2$ has a Hamilton cycle as long as each of its 3-cycles bounds a triangle.

I'm wondering if this extends to higher dimensions:

Prove that for every triangulation of $\mathbb{S}^3$, the 2-skeleton contains a 'Hamilton sphere', i.e. a homeomorph of $\mathbb{S}^2$ containing all vertices, unless it contains certain substructures (which are for you to find).