  The eigenstate thermalization hypothesis describes how most isolated
many-body quantum systems reach thermal equilibrium. However, the hypothesis is
violated by phenomena such as many-body localization and quantum many-body
scars. In this work, we study a finite, two-dimensional, disordered model
hosting a tower of scar states. This construction is a particular instance of a
general framework and we demonstrate its generality by constructing two
disordered models hosting a different tower of scar states. At weak disorder,
we find numerically that the spectra are nonthermal, and the scar states appear
as exact eigenstates with high entropy for certain bipartitions. At strong
disorder, the spectra localize and the scar states are identified as inverted
scars since the scar states are embedded in a localized background as opposed
to a thermal background. We argue that, for the considered type of models, the
localization is stronger than what would be naively expected, and we show this
explicitly for one of the models. The argument also provides guidelines for
obtaining similarly strong localization in other scarred models. We study the
transition from the thermal phase to localization by observing the adjacent gap
ratio shifting from the Wigner surmise to the Poisson distribution with
increasing disorder strength. Moreover, the entanglement entropy transitions
from volume-law scaling with system size at weak disorder to area-law scaling
at strong disorder. Finally, we demonstrate that localization protects scar
revivals for initial states with partial support in the scar subspace.
