All intermediate logics with extra axioms in one variable, except eight, are not strongly omega-complete

In [8] it is proved that all the intermediate logics axiomatizable by formulas in one variable, except four of them, are not strongly complete. We considerably improve this result by showing that all the intermediate logics axiomatizable by formulas in one variable, except eight of them, are not strongly $\omega$-complete. Thus, a definitive classification of such logics with respect to the notions of canonicity, strong completeness, $\omega$-canonicity and strong $\omega$-completeness is given.