Does every variety of locally finite closure algebras have a unification type?

Question

An algebra has one of four unification types - unitary, finitary, infinitary or nullary. A variety $\mathbf{V}$ has type unary, if every member has unary type, finitary, if every member has finitary type, infinitary, if every member has infinitary type, and nullary, if there exists an algebra in $\mathbf{V}$ with nullary type, see, for example, Ghilardi, S. (1997). Unification through projectivity. J. Logic Comput., 7(6):733–752. My question is whether every $\mathbf{V}$ has a unification type, or if it is possible for $\mathbf{V}$ whose type is not nullary to contain algebras of different type. In particular, can this happen for locally finite varteties of closure algebras?