Smooth globally PLI functions are nonlinear least-squares, and so are their gradient-dominated cousins

Boumal, Criscitiello and Rebjock (BCR) in [arXiv:2604.07972, 2026] proved that a smooth real-valued function f defined on a contractible complete Riemannian manifold which satisfies the (global) Polyak-Lojasiewicz inequality (PLI) is necessarily of the form f = f* + phi^2, with phi a submersion, together with a long list of consequences. That hypothesis is unfortunately not available in several problems where one most wants it, among them continuous-time LQR policy optimization and logistic regression, and this is what motivated the hierarchy of generalized PLI inequalities developed several papers. We observe here that the global PLI inequality is stronger than what the proofs in the BCR paper require. Replacing it by a positive definite function, bounded below by a multiple of sqrt near the origin, which is to say, replacing gl-PLI by "sgl-PLI" leaves every structural conclusion of intact. Both immediate weakenings fail, with elementary counterexamples.The conclusions do apply to the continuous-time LQR problem and to logistic regression.