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I was under the impression this version doesn’t need exotic matter?

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.

Ex: https://arxiv.org/abs/2201.00652

They also say: "We present the first general model for subliminal positive-energy, spherically symmetric warp drives"

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.

My understanding is that physics allows you to create a bubble of spacetime without negative energy, but that bubble doesn't accelerate through space.

The ship drags the bubble along with it at sublight speed using normal thrusters.

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