  In a recent work, we numerically studied the radiative properties of the
reverberation phase of pulsar wind nebulae (PWNe), i.e., when the reverse shock
created by the supernova explosion travels back towards the pulsar, compressing
the wind bubble. We focused on several well-characterized PWNe and used them as
examples for introducing the concept of superefficiency. The latter is a period
of the PWN evolution, happening within reverberation, where the luminosity in a
given band exceeds the spin-down power at the time. Here, we explore a broad
range of PWN models to study their reverberation and superefficiency phases in
a systematic way. Armed with these models we consider two aspects: On the one
hand, we analyze via Monte Carlo simulations how many Galactic PWNe are
expected to be reverberating or in a superefficiency stage at any given time,
providing the first such estimations. On the other hand, we focus on searching
for observational signatures of such periods. We analyze archival observations
and check for the existence of possible candidates for superefficient PWNe. We
also provide predictions for the future evolution of the magnetar nebula
J1834.9-0846 (which we consider to be starting its reverberation period) along
the next 50 years. Using our simulations as input we study how sensitive
current and future X-ray satellites (like eXTP or Athena) will be to observe
such evolution, concluding that they will be able to track it in detail.
