We study the N = 4 SYM stress tensor multiplet 4-point function for any value of the complexified coupling tau, and in principle any gauge group (we focus on SU(2) and SU(3) for simplicity). By combining non-perturbative constraints from the numerical bootstrap with two exact constraints from supersymmetric localization, we are able to compute upper bounds on low-lying CFT data (e.g. the Konishi) for any value of tau. These upper bounds are very close to the 4-loop weak coupling predictions in the appropriate regime. We also give preliminary evidence that these upper bounds become small islands under reasonable assumptions, in which case our method would provide a numerical solution to N = 4 SYM for any gauge group and tau. -------- Part of the London Integrability Journal Club. Please register at integrability-london.weebly.com if you are a new participant.