Taking as a starting point the variation in introspective judgments on embedded gapping in English in the literature, the main goal of this paper is to test the ‘No Embedding Constraint’ experimentally. Building on a first experimental study designed to measure the interaction between that-omission and factivity in English embedded complement clauses, we conducted two experiments testing the role of the complementizer in embedded gapping, paying special attention to the semantic nature of the matrix predicates (non-factives vs semi-factives vs true factives). Our results show, on the one hand, that the ‘No Embedding Constraint’ makes too strong claims that are not backed up by our experimental findings, and, on the other hand, that embedded gapping is affected by both the presence/absence of that and by the semantic class of the matrix predicate in English. In particular, embedded gapping seems to be more acceptable under non-factive verbs, especially in the absence of a complementizer. Both constraints (that-omission and factivity) can be accounted for by a constructionist fragment-based analysis, where the gapped clause is a non-finite phrase that has to address the same Question Under Discussion as its source. This explains, in turn, why embedded gapping under true factive predicates is considered significantly less acceptable. We show that the acceptable cases of embedded gapping involve true syntactic embedding (so, the matrix clause has no parenthetical use). We conclude that English has the same sensitivity to the semantic class of the matrix predicate as other languages, but that the requirements on the presence/absence of that are English specific.