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Jean Joseph sent an email to the HoTT mailing list claiming to have defined multiplication for surreals in HoTT.
I haven’t read through the paper yet.
If I have skimmed correctly it seems the author claims that the grothendieck construction on positive surreals is equivalent to the surreals. Then the ring structure on positive surreals gives the ring structure on surreals. This paper is very strangely written, but I can’t say that is a basis for incorrectness.
I don’t think it is correct. (In particular, he claims that the order on surreals is cotransitive, which I believe is certainly false.) I’ve contacted him privately in hopes that he will be able to fix the problems; hopefully updates will be forthcoming.
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