Anantharam, Jog, and Nair recently unified the Shannon–Stam inequality and the entropic form of the Brascamp–Lieb inequalities under a common inequality. They left open the problems of extremizability and characterization of extremizers. Both questions are resolved in the present paper. Separately, we show that the Anantharam–Jog–Nair inequality may be derived as a corollary of the Brascamp–Lieb inequalities, establishing a formal equivalence between the two.