Countable sequentially Ascoli but not Ascoli (space/group)

Determine whether there exists a countable topological space or a countable topological group X that is sequentially Ascoli but not Ascoli.

Background

Sequentially Ascoli spaces are a weakening of the Ascoli property tailored to sequences in Ck(X), and the paper develops extensive relationships between new and classical compact-type classes. The authors show that for countable spaces, κ-sequential, weakly open-compact attainable, and sequentially Ascoli properties coincide (Corollary 2.19(i)), but this does not address whether sequentially Ascoli implies Ascoli in the countable field.

Known examples distinguish sequentially Ascoli from Ascoli (e.g., non-discrete P-spaces are sequentially Ascoli but not Ascoli), yet these are not countable. The problem therefore targets the countable case to determine whether the separation between sequentially Ascoli and Ascoli persists at minimal cardinality, or whether countability forces them to coincide.