Necessity of omega-limit point compactness outside the T1 setting

Determine whether $\omega$-limit point compactness is necessary for every non-$T_1$ topological space to have finite cliques.

Background

The paper proves that $\omega$-limit point compactness, equivalently countable compactness in the authors’ general setting, suffices for the finite clique property. For non-$T_1$ spaces, however, the converse is not established.

The remark notes that non-$T_1$ limit-point-compact spaces can have infinite cliques, so the unresolved issue concerns whether the stronger $\omega$-limit point compactness condition is forced by finite cliques in the general, non-$T_1$ case.