In 2019 Zbigniew Osiak published a paper saying that Einstein's formulas for energy in special relativity are wrong. The (rest) mass-energy equivalency law, E_0 = m c^2 (where c is the speed of light) according to Einstein should be E_0 = m c^2 /2 , Osiak found. Also, the general energy formula (per Einstein E = g m c^2, where g=sqrt(1/(1 - (v/c)^2)) with v the particle velocity, is the "Lorentz factor", that makes it impossible to accelerate an object to c, the speed of light) should be E = g^2 m c^2 /2. So, in addition to a factor of a half, like the rest mass energy, the general energy formula has an extra Lorentz factor, that approaches infinity as a, say, elementary particle's velocity approaches the speed of light. So if Osiak is correct, the very early universe , which consists of a lot of relativistic elementary particles (for a short time as a quark-gluon plasma) has a total energy that is greater than expected according to Einstein relativity by a truly huge factor. So it seems reasonable that such a huge amount of extra energy density might lead to star and galaxy formation more rapidly than the Einstein-relativity based cosmological models predict.
So is Osiak relativity correct? How could such mistakes in special relativity have gone overlooked for over a hundred years? The answer is, the errors weren't overlooked. They were made deliberately because the correct formulas can be easily shown to violate energy conservation. A physical law that violated energy conservation was unacceptable to Einstein (and many co-workers such as Tolman and (later) Rindler and Penrose). However, the energy nonconservation is actually nearly impossible to measure because it occurs only under very extreme conditions such as in the early universe. Also, the quantity P_0 = g m c^2 is a conserved quantity (the temporal component of four-momentum) that is also important in Osiak relativity. The Osiak kinetic energy is actually equal to the classical formula E = p^2 / 2m, where p = g m v is the (relativistic) momentum. The Einstein form of kinetic energy only reduces to the classical form in the classical (low velocity) limit.
Until fairly recently, the idea of energy non-conservation was unthinkable to most or all physicists. After dark energy was proposed, however, some quantum gravity guys proposed energy non-conservation as an explanation for dark energy. All this was prior to the Webb launch but the paper I read was after Osiak published, still they do not seem to be aware of it.
In basic inflationary cosmology, gravitational potential energy from an unknown source (Guth thought originally the Higgs field but it didn't pan out) causes cosmic inflation. It's easy to suppose that energy nonconservation in special relativity can manifest in general relativity as gravitational potential energy. This would be a difficult thing to prove observationally, but if a modified cosmological standard model incorporating Osiak relativity can be constructed, it can be compared with the Webb results, and the matter of whether Osiak relativity is correct can be settled.
I think there is an easier way, though, to prove Osiak is correct. Not trivial, but not cosmological. It involves an old idea of Stuckelberg, that Richard Feynman was fond of, that antimatter particles move backward in time. Osiak relativity also predicts this in a much more straightforward way than in Einstein relativity. I think it could be observable directly.