Suitable sets in linearly orderable topological groups

Determine whether every linearly orderable topological group admits a suitable set, and in particular whether it admits a closed suitable set.

Background

A linearly orderable topological group is a group whose topology is the order topology induced by a linear order compatible with the group structure. The existence of suitable sets in such groups has been a long-standing question.

The paper provides partial answers and related results, including metrizability criteria and consequences for suitable sets under additional hypotheses (e.g., having an ω^ω-base).