There is a hypothesis that a non-self-centric radially-maximal graph of radius \( r \) has at least \( 3r – 1 \) vertices. Moreover, if it has exactly \( 3r – 1 \) vertices, then it is planar with minimum degree \( 1 \) and maximum degree \( 3 \). Using an enhanced exhaustive computer search, we prove this hypothesis for \( r = 4, 5 \).