Computational complexity of unknotting problem?

Question

The Wikipedia article on the unknotting problem says "a major unresolved challenge is to determine [...] whether the problem lies in the complexity class P". It mentions some work towards this result and a recent claim that it is true. But then finally it says "the unknotting problem has the same computational complexity as testing whether an embedding of an undirected graph in Euclidean space is linkless," referencing this paper. Doesn't that statement imply that the unknotting problem is in P, by the graph minor theorem and the existence of a polynomial time algorithm to test for a specific minor in a given graph? What am I missing here? What is the current status of this problem?