I know they are studying the magnetic fields of these stars but...technically, it is gravity that locks one star’s movements to another. Not magnetism.
A star's magnetic field does have some mechanical effects on anything nearby that is either ionized (like interstellar gas) or conducts electricity (like the plasma in a nearby star). Mechanical magnetic effects on the rotation speed of a star are minute, but measurable.
For example, the rotating magnetic field of a spinning neutron star can thrash electrons in the interstellar medium so hard they are accelerated to relativistic speeds and give off visible synchrotron radiation (like the Crab Nebula). This transfer of mechanical energy causes the spinning star to slow down.
If a star is rotating within another star's magnetic field, it will encounter some resistance due to magnetic braking, though gravitational tidal forces are probably vastly more significant. On a smaller scale, convection currents in the Sun are impeded when they are forced to move across magnetic field lines; and this is why sunspots are cooler than the surrounding regions. (The rotating disk in a typical power meter, which passes through a pair of damping magnets, is another illustration of this principle.)
Could be both. Or the magnetic effect might be trying to push them apart, but can't overcome the gravity force, although modifying its effect. Or the magnetic effect could be adding to the gravitational one, again modifying it's effect. Force is force, regardless of it's origin.