The puzzle applies only to convex polyhedra.

The article says:

> The full menagerie of shapes is too diverse to get a handle on, so mathematicians tend to focus on convex polyhedra

The phrase "tend to focus on" suggests it's not an exclusive thing. However, you're right -- it appears that the Rupert property only applies to convex polyhedra, so the article title and text is at the very least incomplete given that a sphere is a shape.

A sphere is not a convex polyhedron

At the limit of faces they are.

Sure, and pi is the limit of a sequence of rational numbers, but lots of properties that hold for rational numbers don't hold for pi.

As you approach sphere you lose Rupertness.

Limiting behaviour can be counterintuitive. As you add vertices to a polyhedron, some properties approach those of a sphere (volume, surface area), but others just get further and further away (number of surface discontinuities). It's not at all obvious which way "Rupertness" will go, or even whether it's monotone with respect to vertex addition.

A sphere has no faces so it's not a convex poloyhedron.

Correction: a sphere has infinite faces so it's not an "convex poloyhedron [sic]." A convex polyhedron must have finite faces, so apeirotopes aren't allowed.

A sphere has no faces, not "infinite" faces.

Convex polyhedra are required to be finite polytopes.