Sticky Existential Rules and Disjunction are Incompatible

Stickiness is one of the well-known properties in the literature that guarantees decidability of query answering under sets of existential rules, that is, Datalog rules extended with existential quantification in rule heads. In this note, we investigate whether this remains true in the case when rule heads are allowed to be disjunctive. We answer this question in the negative, providing a strong undecidability result that shows that the concept of stickiness cannot be extended to disjunctive existential rules, even when considering only fixed atomic queries and a fixed set of rules. This provides evidence that, in order to keep query answering decidable, a stronger property than stickiness is needed in the disjunctive case.