Adequate domains that are not good-factorization domains

Construct an adequate domain that does not have good factorization, or determine that every adequate domain has good factorization.

Background

The paper records the implication from good factorization to adequacy and notes that it has not found an example of an adequate domain without good factorization. Establishing or refuting the converse implication would clarify the relationship between these two divisibility properties.

References

Unrelated to lcm splitting, but still interesting: Are there adequate domains that are not gf?

— LCM Splittings in GCD Domains  (2610.23799 - Preu, 16 Oct 2026) in Section \ref{sec:final}, Final remarks and open questions

Could one reformulate gf and/or adequate in purely topological terms of spectral spaces/trees?

— LCM Splittings in GCD Domains  (2610.23799 - Preu, 16 Oct 2026) in Section \ref{sec:final}, Final remarks and open questions

In general, what are the possible combinations of properties that (pairs of) elements can have in B{e}zout domains?

— LCM Splittings in GCD Domains  (2610.23799 - Preu, 16 Oct 2026) in Section \ref{sec:final}, Final remarks and open questions