The Gaussian stationary point in an inequality motivated by the Z-interference channel was recently conjectured by Costa, Nair, Ng, and Wang to be the global optimizer, which, if true, would imply the optimality of the Han-Kobayashi region for the Gaussian Z-interference channel. This conjecture was known to be true for some parameter regimes, but the validity for all parameters, although suggested by Gaussian tensorization, was previously open. In this paper we construct several counterexamples showing that this conjecture may fail in certain regimes: A simple construction without Hermite polynomial perturbation is proposed, where distributions far from Gaussian are analytically shown to be better than the Gaussian stationary point. As alternatives, we consider perturbation along geodesics under either the $L^2$ or Wasserstein-2 metric, showing that the Gaussian stationary point is unstable in a certain regime. Similarity to stability of the Levy-Cramer theorem is discussed. The stability phase transition point admits a simple characterization in terms of the maximum eigenvalue of the Gaussian maximizer. Similar to the Holley-Stroock principle, we can show that in the stable regime the Gaussian stationary point is optimal in a neighborhood under the $L^{\infty}$-norm with respect to the Gaussian measure. For protocols with constant power control, our counterexamples imply Gaussian suboptimality for the Han-Kobayashi region. Allowing variable power control, we show that the Gaussian optimizers for the Han-Kobayashi region always lie in the stable regime. We propose an amended conjecture, whose validity would imply Gaussian optimality of the Han-Kobayashi bound in a certain regime.