We perform an in-depth complexity analysis of query answering under guarded-based classes of disjunctive tuple-generating dependencies (DTGDs), focusing on (unions of) conjunctive queries ((U)CQs). We show that the problem under investigation is very hard, namely 2E<scp>xp</scp>T<scp>ime</scp>-complete, even for fixed sets of dependencies of a very restricted form. This is a surprising lower bound that demonstrates the enormous impact of disjunction on query answering under guarded-based tuple-generating dependencies, and also reveals the source of complexity for expressive logics such as the guarded fragment of first-order logic. We then proceed to investigate whether prominent subclasses of (U)CQs (i.e., queries of bounded treewidth and hypertree-width, and acyclic queries) have a positive impact on the complexity of the problem under consideration. We show that queries of bounded treewidth and bounded hypertree-width do not reduce the complexity of our problem, even if we focus on predicates of bounded arity or on fixed sets of DTGDs. Regarding acyclic queries, although the problem remains 2E<scp>xp</scp>T<scp>ime</scp>-complete in general, in some relevant settings the complexity reduces to E<scp>xp</scp>T<scp>ime</scp>-complete. Finally, with the aim of identifying tractable cases, we focus our attention on atomic queries. We show that atomic queries do not make the query answering problem easier under classes of guarded-based DTGDs that allow more than one atom to occur in the body of the dependencies. However, the complexity significantly decreases in the case of dependencies that can have only one atom in the body. In particular, we obtain a P<scp>time</scp>-completeness if we focus on predicates of bounded arity, and <i>AC<sub>0</sub></i>-membership when the set of dependencies and the query are fixed. Interestingly, our results can be used as a generic tool for establishing complexity results for query answering under various description logics.