  Eigenstate thermalization hypothesis (ETH) represents a breakthrough in
many-body physics since it allows to link thermalization of physical
observables with the applicability of random matrix theory (RMT). Recent years
were also extremely fruitful in exploring possible counterexamples to
thermalization, ranging, among others, from integrability, single-particle
chaos, many-body localization, many-body scars, to Hilbert-space fragmentation.
In all these cases the conventional ETH is violated. However, it remains
elusive how the conventional ETH breaks down when one approaches the boundaries
of ergodicity, and whether the range of validity of the conventional ETH
coincides with the validity of RMT-like spectral statistics. Here we bridge
this gap and we introduce a scenario of the ETH breakdown in many-body quantum
systems, dubbed fading ergodicity regime, which establishes a link between the
conventional ETH and non-ergodic behavior. We conjecture this scenario to be
relevant for the description of finite many-body systems at the boundaries of
ergodicity, and we provide numerical and analytical arguments for its validity
in the quantum sun model of ergodicity breaking phase transition. For the
latter, we provide evidence that the breakdown of the conventional ETH is not
associated with the breakdown of the RMT-like spectral statistics.
