Countable κ-sequential but not sequential (space/group)

Determine whether there exists a countable topological space or a countable topological group X such that X is κ-sequential but X is not sequential.

Background

The paper introduces and studies new compact-type classes (open-compact attainable, weakly open-compact attainable, and κ-sequential) and relates them to classical notions such as sequential and Fréchet–Urysohn spaces, Ascoli and sequentially Ascoli spaces, k′-, kR-, and sR-spaces. Within this framework, several structural results and counterexamples are provided.

For countable spaces, the authors establish strong equivalences: X is κ-sequential if and only if X is weakly open-compact attainable if and only if X is sequentially Ascoli (Corollary 2.19(i)). However, these results do not settle whether κ-sequentiality collapses to sequentiality in the countable setting. This problem asks specifically whether a countable space or group can be κ-sequential without being sequential, thereby probing the boundary between these properties in the minimal-cardinality case.