Is magnetic field conservative in nature?
Magnetic field lines do go in closed paths but that's not the definition of conservative.
Rather, a field is conservative when the force on a test particle moving around any closed path does no net work. But magnetic fields only act on moving charges, and at right angles to the motion, so the work is always zero and the concept doesn't properly apply.
Also, if there were magnetic monopoles, they would try to follow the magnetic field the way electric charges try to follow the electric field lines.
Now we consider the magnetic field due to magnetic monopoles, which is going to be conservative.
So Is there a precise answer? or the nature of the magnetic field depends on the way you produce it? If the latter is the case then, is the magnetic field created by monopoles conservative?