Previous research has uncovered that, despite the omnipresence of focus in utterances, children typically do not compute the exhaustivity inference associated with cleft(-like) syntactic focus constructions at adult-like levels before 7 years of age. Children's comparable limitations with lexically triggered scalar implicatures, inferences with an essentially identical logical structure, have been argued to stem not from the lack of their competence to compute the implicature itself, but from their immature contextualization abilities to identify relevant scalar alternatives. We address the question whether the same extraneous factor may underlie preschooler's difficulties with focus exhaustification in a comprehension study of 5- to 6-year-old Hungarian children which investigated the effect of contextual cues on their focus interpretation. It was found that while the presence of an explicit Question Under Discussion significantly raises children's accuracy in identifying the focus and its alternatives (sub-experiment 1), it fails to induce a similar increase in the proportion of exhaustive interpretations (sub-experiment 2). These results indicate that, in contrast to the case of scalar implicatures, children's low rate of exhaustification of focus reflects a deeper, less context-dependent difference from adult-like comprehension, possibly rooted in the linguistic representation of focus exhaustivity.