FWIW I'm not buying her argument that the comoving frame represents a preferred frame in SR either. She glosses over quite some fundamental questions there. I'd love to see this written up as a proper paper.
ds^2 = c * dt^2 - dx^2 - dy^2 - dz^2
If ds^2 > 0 then the interval is timelike.
If ds^2 = 0 then the interval is lightlike.
If ds^2 < 0 then the interval is spacelike.
Let's just focus on an object moving strictly along the x-axis faster than the speed of light, this results in a spacetime interval as follows:
ds^2 = c * dt^2 - dx^2
Since the object is moving faster than light, the dx^2 term would exceed the c * dt^2 term, resulting in a ds^2 < 0 and hence spacelike rather than timelike curve.
Within the context of special relativity, an object travelling faster than light would be travelling along a spacelike curve, not a timelike curve. The significance of general relativity is that there are solutions that allow for the trajectory of an object to move along a closed timelike curve without exceeding the speed of light.
The whole point of CTCs is that in exotic spacetime geometries, in particular wormhole-like structures, there can be curves that are entirely timelike, meaning that the object never moves faster than light locally, and are closed at the same time, meaning the object ends up at the same point in time from where it started.
In SR, a closed curve requires at least some spacelike sections of the curve on which the object moves faster than light.