Conjecture on Carathéodory numbers for strong convexity and subsets of facet normals

Let $K$ be a polytope with set of facet normals $H$, and let $h(H)$ denote the Carathéodory number for $H$-convexity. If the Carathéodory number for $K$-strong convexity is $k$, then

$$h(H)\le k\le \max_{H'\subseteq H} h(H').$$

Subset-normal bound conjecture. The inequalities above hold.

The lower bound is established in the paper, while the upper bound is presented as a broader conjecture; the proposed characterization of the one-less-than-the-number-of-facets case is described as a consequence of it.