I should have mentioned that the sites I quoted were dug up by the good people in the Deep in Japan FB group.

Thanks. Meikyo has 語源未詳 too. Perhaps we could add ("product name"), or just leave it alone.

can't find any evidence of a babybjörn product outside of Japan called stye/sty/staj and I'm not aware od any Swedish word pronounced like that either. the theory doesn't strike me as likely because it appears to be very recently coined - I found no relevant 20th century hits on google books - and if it were originallh a product name then surely it'd be trademarked abd easily traceable.

https://news.livedoor.com/article/detail/8524989/
赤ちゃんのよだれかけを日本では「スタイ」と呼びますが、これはいったい何語なんでしょう?　実はこれはスウェーデンの大手ベビー用品メーカーのベイビービヨルン社のプラスティック製よだれかけの商品名で、これが日本で大ヒットしたことから「スタイ」の名が定着したようです。英語では「Bib」といいます。「sty」は「豚小屋」または「ものもらい」の意味になるので、間違っても使わないようにしましょう。
Also: https://www.psling.net/bib/

So, should we have Swedish as the lsrc? Over to Marcus et al.

When I dug deeper they were mostly Japanese items being sold. Very odd.