A substantial portion of distributed computing research is dedicated to terminating problems like consensus and similar agreement problems. However, non-terminating problems have been intensively studied in the context of self-stabilizing distributed algorithms, where processes may start from arbitrary initial states and can tolerate arbitrary transient faults. In between lie stabilizing problems, where the processes start from a well-defined initial state, but do not need to decide irrevocably and are allowed to change their decision finitely often until a stable decision is eventually reached. In this paper, we introduce the novel Delayed Lossy-Link (DLL) model, and the Lossy Iterated Immediate Snapshot Model (LIIS), for which we show stabilizing consensus to be impossible. The DLL model is introduced as a variant of the well-known Lossy-Link model, which admits silence periods of arbitrary but finite length. The LIIS model is a variant of the Iterated Immediate Snapshot (IIS), model which admits finite length periods of at most $f$ omission faults per layer. In particular, we show that stabilizing consensus is impossible even when $f=1$. Our results show that even in a model with very strong connectivity, namely, the Iterated Immediate Snapshot (IIS) model, a single omission fault per layer effectively disables stabilizing consensus. Furthermore, since the DLL model always has a perpetual broadcaster, the mere existence of a perpetual broadcaster, even in a crash-free setting, is not sufficient for solving stabilizing consensus, negatively answering the open question posed by Charron-Bost and Moran.