Wouldn't that allow for a "perpetual motion" setup where a blob of normal mass is in front of a blob of negative mass? The negative mass would push the positive mass forward, and then be attracted to it and follow. What am I misunderstanding?
For an easier to visualize example imagine placing one magnet positive pole up inside of a tube, and then you place another magnet positive pole down in the same tube. It would create a similar effect. There is a repulsive force being perfectly balanced out by an attractive force and the two reach an equilibrium where the magnet on top is 'floating' with the net of all forces in this closed system being 0.
It still goes away but the acceleration is in the same direction. It is weird and easy to be confused.
An idealized -M + M = 0 pair of particles (one normal, one negative mass) could experience runaway acceleration. Author touches on this in the paper as a potential source for the incredibly high-energy extra-galactic cosmic rays.
Granted, the paper describes a model in which something that acts like a particle with negative mass exerts this force upon things, but has no claims as to what a physical manifestation of this would look like.