We show log-supermodularity of densities on a Euclidean space is preserved under convolution with log-concave product densities, confirming a generalized form of a conjecture of Zartash and Robeva. Additionally, this stability yields a "conditional entropy power inequality" for log-supermodular random vectors. The proof relies on a known continuous form of the Ahlswede-Daykin four-function theorem, for which we give a new proof. The talk is based on joint work with James Melbourne and Cyril Roberto.