We study four-point correlation functions of the stress-tensor multiplet in \mathcal{N} = 4 super Yang-Mills (sYM) theory by leveraging integrability and localization techniques. We combine dispersive sum rules and spectral information from integrability, used previously, with integrated constraints from supersymmetric localization. We obtain two-sided bounds on the OPE coefficient of the so-called Konishi operator in the planar limit at any value of the 't Hooft coupling ranging from weak to strong coupling. In addition to individual OPE coefficients, we discuss how to bound the correlation function itself and obtain two-sided bounds at various values of the cross-ratios and coupling. Lastly, considering the limit of large 't Hooft coupling, we connect the analysis with that of an analogous flat space problem involving the Virasoro-Shapiro amplitude.
