The first thought that comes to my mind on reading this is the fact that through this, we are making a current source which derives energy out of itself, and has a seemingly perpetual life. This is what makes me feel something is wrong.
So let’s assume that the process of making the magnetic field constant happens instantaneously. Initally, when the magnetic field is swiched on, it will oppose its increasing flux. Now when you will make it constant, basically the source of induced current dies,and the current starts its process of decrement. Just when it decreases, the flux through the ring by its own current decreases, so it generates some current in the same direction to increase its flux.
But at this point in time, the value of current is smaller than its initial value and since current induced by self induction in this case is proportional to the rate of change of current in the circuit, for approximately similar time intervals, a smaller current will have a smaller change(w.r.t Zero). This would mean that the induced current will not be able to make up for the intial decrememt.
This process will only magnify itself as time passes by and the curcuit will finally drop to almost zero.
Just like in discharging events it takes an infinite time to reach that absolute zero level, it possibly will be the same case here, but any observable current will die out almost instantaneously.
Also in any case a wire will have atleast some amount of resistance, which will lead to energy loss.