User blog:Triakula/FGH and SGH at large ordinals

Can \(g_{\omega_1^\text{CK}}\) eventually outgrow \(f_{\omega_1^\text{CK}}\) if fundamental sequences are properly given? If yes, are there known variants of it?

Generally, is it true that initially faster hierarchies imply that they are always faster at all ordinals regardless of fundamental sequences?