Conjecture on the facet-normal characterization of near-maximal strong Carathéodory number

Let $K$ be a polytope, and let its normals be the normal vectors defining its facets. The Carathéodory number for $K$-strong convexity is one less than the number of facets of $K$.

Facet-normal characterization conjecture. The normals of $K$ are the same as either those of a simplex bounded by an extra facet, or those of a pyramid that is not a simplex.

This would characterize the case in which the strong-convexity Carathéodory number is one less than the number of facets; the converse is established by the corresponding examples for pyramids and simplices bounded by an extra facet.