Two types of filtrations for $\mathrm{w}\mathrm{K}4$ and its relatives

We study the finite model property of subframe logics with expressible transitive reflexive closure modality. For $m>0$, let ${\mathrm{L}}_m$ be the logic defined by axiom ${\mathrm{◊}}^{m+1}p\to \mathrm{◊}p\vee p$. We construct filtrations for the logics ${\mathrm{L}}_m$. It follows that these logics and their tense counterparts have the finite model property. Then we show that every canonical subframe logic that contains ${\mathrm{L}}_m$ have the finite model property.