The core chase, a popular algorithm for answering conjunctive queries (CQs) over existential rules, is guaranteed to terminate and compute a finite universal model whenever one exists, leading to the equivalence of the universal-model-based and the chase-based definitions of finite expansion sets (fes)-a class of rulesets featuring decidable CQ entailment. In case of non-Termination, however, it is non-Trivial to define a ''result'' of the core chase, due to its non-monotonicity. This causes complications when dealing with advanced decidability criteria based on the existence of (universal) models of finite treewidth. For these, sufficient chase-based conditions have only been established for weaker, monotonic chase variants. This paper investigates the-prima facie plausible-hypothesis that the existence of a treewidth-bounded universal model and the existence of a treewidth-bounded core-chase sequence coincide-which would conveniently entail decidable CQ entailment whenever the latter holds. Perhaps surprisingly, carefully crafted examples show that both directions of this hypothesized correspondence fail. On a positive note, we are still able to define an aggregation scheme for the infinite core chase that preserves treewidth bounds and produces a finitely universal model, i.e., one that satisfies exactly the entailed CQs. This allows us to prove that the existence of a treewidth-bounded core-chase sequence does warrant decidability of CQ entailment (yet, on other grounds than expected). Hence, for the first time, we are able to define a chase-based notion of bounded treewidth sets of rules that subsumes fes.