Amended Gaussian optimality conjecture for the Han–Kobayashi region

Let $(N_1,N_2,u,q_1,q_2)$ denote positive channel and power parameters, and let the isotropic condition mean that the maximum in the stated theorem is achieved by isotropic Gaussian inputs for every positive integer dimension. Let 1-letter HKGS without power control and 1-letter HKGS with power control denote the corresponding Gaussian Han–Kobayashi schemes. Amended Gaussian optimality conjecture. The set of $(N_1,N_2,u,q_1,q_2)$ satisfying the isotropic condition in the theorem equals the set of parameters for which 1-letter HKGS without power control matches 1-letter HKGS with power control. This conjecture isolates the stability regime in which power control is unnecessary for the Gaussian Han–Kobayashi bound. The paper states that the isotropic parameter set is nonempty and that a more precise characterization remains open; its status evidence says that disproving this conjecture would disprove optimality of HKGS with power control for the whole capacity region.