Soundness and strong completeness of ${\bf K^W}$ for filters and $(mc)$-frames

Let ${\bf K^W}$ be the logic considered in the paper, let a filter be a neighborhood frame whose neighborhood collections satisfy the filter conditions defined in the surrounding development, and let an $(mc)$-frame be a neighborhood frame satisfying condition $(mc)$. The completeness conjecture. ${\bf K^W}$ is sound and strongly complete with respect to the class of filters, and also with respect to the class of $(mc)$-frames.

Soundness is stated to be straightforward, whereas the paper explains that the completeness proof is not currently known by the proposed canonical-model argument. The authors note that completeness can instead be obtained from completeness with respect to relational frames, together with pointwise equivalent augmented models and the fact that augmented models are filters; the conjecture is therefore presented as unresolved in the main development.