Such a quasi-polynomial algorithm has just been claimed by Marc Lackenby from Oxford University. He will present in next Tuesday (02 Feb 2021) in a Zoom talk: https://www.math.ucdavis.edu/research/seminars/?talk_id=6082
The outline from Marc Lackenby's talk about a quasipolynomial algorithm for Unknottedness.
Unknot recognition in quasipolynomial time outline.. Under the talks section there are slides about the algorithm.
And for the seminar video, which has been uploaded link to the video and more about the result. .