  Multi-messenger astronomy with gravitational waves is a blooming area whose
limits are not clear. After the first detection of binary black hole merger and
the famous event GW170817 and its electromagnetic counterpart, the
compact-object astrophysical community is starting to grasp the physical
implications of such event and trying to improve numerical models to compare
with future observations. Moreover, recent detections made by the NICER
collaboration increased the tension between several theoretical models used to
describe matter in the inner core of compact objects. In this paper, we focus
on quadrupolar purely spacetime wI-modes of oscillating compact objects
described using a wide range of hybrid equations of state able to include
several theoretical possibilities of exotic matter in the inner core of such
stars. We study the case in which a sharp first order hadron-quark phase
transition occurs and explore the scenarios of rapid and slow phase conversions
at the interface. We put special attention on the validity of universal
relationships for the oscillation frequency and damping time that might help
unravel the mysteries hidden at the inner cores of compact objects. We show
that, within the slow conversion regime where extended branches of hybrid stars
appear, universal relationships for wI-modes proposed in the literature do not
hold.

High Energy Astrophysical Phenomena