I was looking in Ravi Vakil's notes on Intersection Theory, Class 20, where he introduces the bivariant intersection theory, in particular the Chow ring $A^\ast (X)$.

On p. 2, he writes the following which I had no idea about:

Alarming fact:This ring is apparently not known to be commutative in general, because the argument requires resolution of singularities. (It is known to be commutative in characteristic 0, and for smooth things in positive characteristic, and a few more things.) I think it should be possible to show that the ring is commutative in general using technology not available when this theory was first developed, using Johan de Jong’s “alteration theorem” in positive characteristic. If you would like to patch this hole, then come talk to me.

Vakil's notes for this class are from 2004, so plenty could have happened since then. Is this still an open problem?