Polynomial space hardness without disjunction property

In [HT11], Horč́ık and Terui show that if a substructural logic enjoys the disjunction property, then its tautology problem is PSPACE-hard. We prove that all substructural logics in the interval between intuitionistic logic and generalized Hájek basic logic have a PSPACE-hard tautology problem, which implies that uncountably many substructural logics lacking the disjunction property have a PSPACE-hard tautology problem.