Uncountable increasing chains below purity

Determine whether \(\mathscr{L}^{1}_{\mathbb{Z}}\) contains an uncountable strictly increasing chain.

Background

The authors construct a countable strictly increasing chain in LZ1\mathscr{L}^{1}_{\mathbb{Z}} by requiring preservation of divisibility by successive powers of a fixed prime. This demonstrates that nontrivial increasing chains exist below purity.

Whether such a chain can have uncountable cardinality is left unresolved.

References

It is likewise open whether \mathscr{L}{1}_{\mathbb{Z}} contains an uncountable strictly increasing chain.

— The lattice of abstract elementary classes of modules  (2608.19548 - Hyttinen et al., 20 Aug 2026) in Section 5.1, after the proof of the proposition constructing a countable strictly increasing chain