A request has been made: <https://meta.wikimedia.org/wiki/Requests_for_new_languages/Wikipedia_Bajjika>. I think it is an uncontroversial case of eligibility: The language recently received an ISO 639-3 code, vjk, and usually we follow ISO 639-3 anyway.

But I've also read the ISO change request at <https://iso639-3.sil.org/request/2022-004> and the article <https://www.researchgate.net/publication/259569512_The_Bajjika_language_and_speech_community_Abhishek_Kumar_Kashyap> and from them I conclude that Bajjika, previously classified as a dialect of Maithili, is sufficiently different from that and other related languages, so that it is"sufficiently unique that it could not coexist on a more general wiki" [LPP].

As it is currently not clear if the request was made by a (native) speaker of the language, or just "in general", as with past practice, I suggest marking it as "on hold, pending the appearance of native speakers" rather than "eligible", even though it would be eligible per se.