The conjecture is confirmed for families of at most 50 sets and other special cases, and Gilmer's 2022 argument reaches only a weaker constant near 0.38 rather than 0.5, so no proof reaches the full one-half bound for every family.
What would resolve this A proof that some element always belongs to at least half of the sets in any union-closed family, or a counterexample family where no element does.
OpenCombinatoricsUnion-Closed Sets Conjecture (Wikipedia)