Local finiteness of intermediate logics of finite powers of the natural numbers

Let $n$ be finite, and let ${\operatorname{ILog}}(\omega^n,\preceq)$ denote the intermediate logic of the frame $(\omega^n,\preceq)$. Local finiteness claim. For all finite $n$,

$${\operatorname{ILog}}(\omega^n,\preceq)\text{ is locally finite}.$$

This statement appears in the discussion of open problems and conjectures, but the supplied text does not explicitly establish whether it is proved or remains open.