The recent Planck Legacy release confirmed the presence of an enhanced lensing amplitude in the cosmic microwave background (CMB) power spectra. Notably, this amplitude is higher than that estimated by the lambda cold dark matter model (ΛCDM), which endorses a positively curved early Universe with a confidence level greater than 99%. Although General Relativity (GR) performs accurately in the local/present Universe where spacetime is almost flat, its lost boundary term, incompatibility with Quantum Mechanics and the necessity of dark matter/energy could indicate its incompleteness. By utilising the Einstein–Hilbert action, this letter presents extended field equations by considering the pre-existing/background curvature and the boundary contribution. The extended field equations consist of Einstein field equations with a conformal transformation feature in addition to the boundary term, which can remove singularities, satisfy a conformal invariance theory and facilitate its quantisation.