  We consider solutions to the linear wave equation $\Box_{g}\varphi = 0$ on a
class of approximately $k$-self-similar naked singularity interiors. This
equation models the blue-shift effect, an instability exploited by
Christodoulou in the proof of low-regularity weak cosmic censorship. Using a
combination of resonance expansions and multiplier estimates, we find in the
small-mass regime $k^2 \ll 1$ that the asymptotics of solutions are strongly
sensitive to the regularity assumed on outgoing, characteristic initial data
across the past light-cone of the singularity. Above a threshold regularity set
by the $k$-self-similar scalar field, solutions are shown to always obey
self-similar bounds, indicating that the blue-shift instability competes with
the stabilizing influence of high regularity. We conclude that a proper
statement of weak cosmic censorship, as well as an understanding of the role of
naked singularities in phenomena such as critical collapse, may depend on the
topology of initial data.
