This abstract says only that they've reduced the need for exotic matter by some orders of magnitude. Presumably it's still required.
However, there were some recent papers describing warp drive geometries with only positive mass.
However, there were some recent papers describing warp drive geometries with only positive mass.
Not that I understand the physics well enough to do anything with this, but as you say, there is other work that's definitely (claiming) positive-only energy densities.
The ship drags the bubble along with it at sublight speed using normal thrusters.