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Several recent studies have shown that velocity differences of very wide binary stars, measured to high precision with GAIA, can potentially provide an interesting test for modified-gravity theories which attempt to emulate dark matter. These systems should be entirely Newtonian according to standard dark-matter theories, while the predictions for MOND-like theories are distinctly different, if the various observational issues can be overcome. Here we provide an updated version of our 2019 study using the recent GAIA EDR3 data release: we select a large sample of 73 159 candidate wide binary stars with distance <300 parsec and magnitudes G<17 from GAIA EDR3, and estimate component masses using a main-sequence mass-luminosity relation. We then examine the frequency distribution of pairwise relative projected velocity (relative to circular-orbit value) as a function of projected separation, compared to simulations; as before, these distributions show a clear peak at a value close to Newtonian expectations, along with a long 'tail' which extends to much larger velocity ratios and may well be caused by hierarchical triple systems with an unresolved or unseen third star. We then fit these observed distributions with a simulated mixture of binary, triple and flyby populations, for GR or MOND orbits, and find that standard gravity is somewhat preferred over one specific implementation of MOND; though we have not yet explored the full parameter space of triple population models and MOND versions. Improved data from future GAIA releases, and followup of a subset of systems to better characterise the triple population, should allow wide binaries to become a decisive test of GR vs MOND in the future.