Nope, suppose, you do manage to prevent the ring form flying away in the initial phase, after the DC current stablises, the field also stablises. Hence the effect from Lenz law dies away and the ring will no longer felt anything from the coil
However, since aluminium is paramagnetic, then yes you will expect the ring will over time be magnetised to the same direction as the coil, thus when you let go of the ring, it might end up attracted to the coil and refuse to move up (until you switch off the field of course, which then the aluminium will demagnetise slowly afterwards as the spins rel…