As Gaetz's letter to the Architect argued, correctly:

[T]he 117th Congress adjourned sine die, and a Speaker from the 118th Congress has not been elected. After three undeciding votes, no member can lay claim to this office.  
What is the basis in law, House rule, or precedent to allow someone who has placed second in three successive speaker elections to occupy the Speaker of the House Office? How long will he remain there before he is considered a squatter?  
Please write back promptly as it seems Mr. McCarthy can no longer be considered Speaker-Designate following today's balloting.

It seems correct that the Speaker's office ought to be vacant for now and, if not, the person with the strongest claim to it is actually Rep. Hakeem Jeffries, who's come in first in all three Speakership votes so far.