Recently it has been noted by Di Valentino, Melchiorri and Silk (2019) that the enhanced lensing signal relative to that expected in the spatially flat ΛCDM model poses a possible crisis for the Friedmann–Lemaître–Robertson–Walker (FLRW) class of models usually used to interpret cosmological data. The ‘crisis’ amounts to inconsistencies between cosmological datasets arising when the FLRW curvature parameter Ω k0 is determined from the data rather than constrained to be zero a priori. Moreover, the already substantial discrepancy between the Hubble parameter as determined by Planck and local observations increases to the level of 5σ. While such inconsistencies might arise from systematic effects of astrophysical origin affecting the Planck cosmic microwave background (CMB) power spectra at small angular scales, it is an option that the inconsistencies are due to the failure of the FLRW assumption. In this paper we recall how the FLRW curvature ansatz is expected to be violated for generic relativistic spacetimes. We explain how the FLRW conservation equation for volume-averaged spatial curvature is modified through structure formation, and we illustrate in a simple framework how the curvature tension in a FLRW spacetime can be resolved—and is even expected to occur—from the point of view of general relativity. Requiring early-time convergence towards a Friedmannian model with a spatial curvature parameter Ω k0 equal to that preferred from the Planck power spectra resolves the Hubble tension within our dark energy-free model.