  In this follow up work to [45, 33, 32, 46] we introduce and study a notion of
geodesic stability restricted to rays with prescribed singularity types. A
number of notions of interest fit into this framework, in particular algebraic-
and transcendental K-polystability, equivariant K-polystability, and the
geodesic K-polystability notion introduced by the author in [46]. We provide a
partial comparison of the above notions, and show equivalence of some of these
notions provided that the underlying manifold satisfies a certain 'weak cscK'
condition. As an application this proves K-polystability of a new family of
cscK manifolds with irrational polarization.
