Even after a current, if it will continue to rise. If there is a current, this might be comparable to a force pushing the balloon upwards. There is nothing to stop it from moving upwards unless the current stops.
No, upward current is different from buoyant force. An upward current indicates there is movement of something pushing it up whereas this is not characteristic of buoyant force (placing a balloon of air in water will cause the balloon to rise spontaneously; you don't need a current).
Rereading what I wrote, i don't like my explanation 😛. Sorry i'm not great at physics. But I hope it helps a bit.