The primitive relay channel, introduced by Cover in 1987, is the simplest single-source single-destination network model that captures some of the most essential features and challenges of relaying in wireless networks. Recently, Wu and Ozgur developed upper bounds on the capacity of this channel that are tighter than the cutset bound. In this paper, we recover, generalize and improve their upper bounds with simpler proofs that rely on a converse technique recently introduced by Liu, van Handel and Verdú that builds on reverse hypercontractivity. To our knowledge, this is the first application of reverse hypercontractivity for proving first-order converses in network information theory.