The booster dose increases antibody titers significantly above what they were after the second dose (or first in the case of J&J) so someone who is boosted shou

The booster dose increases antibody titers significantly above what they were after the second dose (or first in the case of J&J) so someone who is boosted should be more protected than someone who got their second dose at the same time. Additionally, through the process of affinity maturation, the antibodies you have after a booster will be more likely to recognize Omicron because your immune system has learned to recognize more features of the virus spike protein and is more robust to mutations.