The classification conjecture for immersions with the central cross-cut property

Let $F\colon M^{n}\to\mathbf{R}^{n+1}$ be a complete immersion with the central cross-cut property, meaning that at least one clean cross-cut exists and the image of every clean cross-cut has central symmetry. A central cylinder is an immersion with a cross-cut preserved by a line of translations and by a central reflection; a tubular quadric is a non-degenerate quadric hypersurface affinely equivalent to a locus of the form

$$x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\pm x_{n+1}^{2}=c\in\mathbf{R}.$$

Classification conjecture. Every complete immersion $F\colon M^{n}\to\mathbf{R}^{n+1}$ with the central cross-cut property must be either a central cylinder or a tubular quadric.

In Euclidean three-space, central cylinders and spheres provide the known examples, with tubular quadrics including all non-degenerate quadrics. The conjecture asserts that these two classes exhaust all complete immersions whose clean cross-cuts have central symmetry; the supplied text gives no resolution evidence.