What are the antibody tests showing now as of late june

FWIW another … small … study with roughly consistent results.

[quote]… The latest study provides more solid evidence that there are SARS-CoV-2-specific T cells that are induced by the infection, says Antonio Bertoletti, who studies T cells at Duke-NUS Medical School in Singapore and was not involved in the work. This is important because researchers have put a lot of emphasis on antibodies, he adds, but it’s still not clear whether antibodies or T cells are more important for protection from the virus. …

… de Vries and colleagues collected blood from 10 COVID-19 patients—four women and six men admitted to the intensive care unit with acute respiratory distress syndrome requiring ventilation—and 10 healthy volunteers. The team exposed the study participants’ immune cells to a pool of predicted SARS-CoV-2 peptides in vitro to see whether or not the cells would react. …

The study authors reported that two [of ten] healthy controls also had circulating SARS-CoV-2-fighting T cells, something that showed up in healthy controls in the study of milder cases of COVID-19 in May. “A likely explanation is that this is reflective of exposure to common cold coronaviruses which . . . are cousins of SARS-CoV-2,” Sette says. …
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Bolding mine.

Local paper just told me they are doing that, around here. Antibody test, not the one for current infection. They send you the results, several days later.