Suitable sets in generalized orderable topological groups with an ω^ω-base

Determine whether every generalized orderable topological group (i.e., a topological group whose underlying space is a generalized ordered space) that has an ω^ω-base admits a suitable set.

Background

An ω^ω-base is a structured neighborhood base indexed by functions from ω to ω with the natural partial order. The paper proves metrizability and existence of suitable sets for linearly orderable groups with an ω^ω-base, and raises the more general question for generalized orderable groups.

This is explicitly stated as an unknown question, indicating a broader class beyond linearly orderable groups.