  We employ partitioning methods, in the spirit of Montiel--Ros but here recast
for general actions of compact Lie groups, to prove effective lower bounds on
the Morse index of certain families of closed minimal hypersurfaces in the
round four-dimensional sphere, and of free boundary minimal hypersurfaces in
the Euclidean four-dimensional ball. Our analysis reveals, in particular,
phenomena of linear index growth for sequences of minimal hypersurfaces of
fixed topological type, in strong contrast to the three-dimensional scenario.
