The following has ben found via Wikipedia page “Gravitational interaction of antimatter”.
Another experimental test has been provided by the supernova SN1987a (anti)neutrinos, and this has been published in two brief reports in Phys. Rev. D in 1988  and 1989 .
After the explosion of this supernova, 19 antineutrinos have been detected at IMB and Kamiokande on earth within a time wintow of 13s. Statistically, 3 to 4 of the events should be due to neutrinos instead of antineutrinos. If at least one of the 19 event involves a neutrino, which according to , had a probability higher than 90% (between 92% and 99%), then the fact that all events happened simultaneously tells us that they felt the same gravitational field from the Galaxy during their travel. More quantitatively, this field is estimated to have changed the travel time of the neutrino (and the light) by a few months (1 to 6 according to ), allowing to upper bound the “gravitational difference” between neutrinos and anti neutrinos to a few part per million.
I have no idea whether this data has been analyzed since 1989, e.g. taking into account the 5 (anti)neutrino detected in the soviet Baksan Neutrino Observatory. But until another supernova explodes in the (not to close) neighbourhood, we probably won’t get much more experimental information on the relative weight of neutrinos and antineutrinos !
- J. M. LoSecco, “Limits on CP invariance in general relativity”,
Phys. Rev. D 38, 3313 (1988). This paper is very short (half a page !), and easy to understand.
- Sandip Pakvasa, Walter A. Simmons, and Thomas J. Weiler,
“Test of equivalence principle for neutrinos and antineutrinos”,
Phys. Rev. D 39, 1761 (1989)