Where one of the authors says "So in this sense, our work is expected to be the end of the road for this problem, although we don't know yet how to prove this rigorously.”
Though I confess I am not up on the latest happenings in this area!
The author is not saying that they don't know how to prove rigorously that their algorithm is O(n log(n)). They have rigorously proven that.
What they are saying is that they don't know how to prove that you can't do better than O(n log(n)).
Schönhage and Strassen conjectured that O(n log(n)) was the lower limit for this problem, and it seems that most experts believe they were right abut that. Hence, the expectation that with the discovery of an actual O(n log(n)) algorithm we are at the end of the road for this problem.
Where one of the authors says "So in this sense, our work is expected to be the end of the road for this problem, although we don't know yet how to prove this rigorously.”
Though I confess I am not up on the latest happenings in this area!