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The James Webb Space Telescope (JWST), which has recently become operational, is capable of detecting objects at record-breaking redshifts, $z \gtrsim 15$. This is a crucial advance for observational cosmology, as at these redshifts the differences between alternative cosmological models manifest themselves in the most obvious way. In recent years, some observational hints have emerged indicating that the Standard Cosmological Model could require correcting. One of these hints is related to the discovery of remote galaxies whose redshifts correspond to the very young Universe (less than one billion years after the Big Bang) but which are similar to nearby galaxies. The issue is that such galaxies in the early Universe do not have enough time to evolve into something similar to the late-Universe galaxies. JWST observations of high-redshift objects are expected to shed light on the origin of this issue. Here we provide results on performing the ``angular diameter -- redshift'' cosmological test for the first JWST observation data. We compare this result with predictions of the standard $Λ$CDM cosmological model and some static cosmological models, including Zwicky's ``tired-light'' model. The latter is currently assumed to be ruled out by observations. We challenge this assumption and show that a static model can provide a natural and straightforward way of solving the puzzle of the well-evolved galaxies and better agreements with the results of the JWST ``angular diameter -- redshift'' test at higher redshifts than the correcting evolution model within the $Λ$CDM framework. We discuss several cosmological tests that will be important for further research on the possibility of revising the expanding Universe paradigm.