A short-circuited coil maintains its current if it is superconducting. A coil is charged with current in the same way that a capacitor is charged with voltage.
Our capacitors are almost ideal components because leakage currents are extremely low and losses in the dielectrics can be negligible.
Conversely, our coils are very poor components, mainly because of their resistance, which causes a coil charged with current to lose it very quickly.
But even if we had coils as ideal as our capacitors, which would maintain a current for as long as a capacitor maintains a voltage, this would not produce any more free energy than we get with capacitors.
A permanent magnet is ultimately just a set of superconducting nano-coils, which are electron spins, and no one has ever succeeded in using permanent magnets to produce a cycle that would generate work. And for good reason: magnetic work requires a variation in magnetic potential energy. If we don't have it, that's the cause of the effect we're making disappear. We can have it on part of a cycle, for example a magnet attracting a ferromagnetic piece. But to return that piece to its starting point, we would have to expend just as much energy.
Some will say that with a current in a coil, we could cut the current on the second part of the cycle and return the ferromagnetic piece to its starting point at zero cost. Wrong! This overlooks the fact that the work done in the first part of the cycle was at the expense of potential magnetic energy, meaning that the energy in the final magnetic field that will be recovered will be less than that initially applied, and the difference will represent the work that has been done.
The idea that a current maintained in a coil could be used to produce free energy is a remake of SMOT. It is equally inept. I understand that people had fun with it at one point, I was no exception, but after 20 to 30 years of failure in this area, it is time to understand the basic principles and realise that changing the DIY parts with coils instead of magnets does not invalidate a principle.
Simpletons see new fields of ‘research’ easy because they have replaced the red insulation on a conductor with green insulation, so clearly, it's not the same thing at all

(I'm hardly exaggerating). They see simplicity everywhere because they do not understand the relationships between the pieces of the physics puzzle, which are beyond them. Moreover, if in two centuries of electromagnetism everyone has failed to produce free energy, assuming that it is possible, it is because producing free energy is surely a subtle matter. They don't even grasp this simple idea; their contempt for scientists and their pretentiousness blind them.
Only the physics principles are (relatively) simple. But for free energy, new principles or new energy sources need to be found. We cannot count on those who see this as easy and go round in circle. If they could work a little, achieve something or think before they speak and utter childish naiveties with the assurance of an omniscient person, we could breathe a sigh of relief.