Visser frames for sublogics of $\mathbf{I}\mathbf{L}$

We study the modal completeness and the finite frame property of several sublogics of the logic $\mathbf{I}\mathbf{L}$ of interpretability with respect to Visser frames, which are also called simplified Veltman frames. Among other things, we prove that the logic $\mathbf{C}\mathbf{L}$ of conservativity has the finite frame property with respect to that frames. This is an affirmative solution to Ignatiev’s problem.