Arnold and Asia are playing 2HG.

Arnold controls a Seedborn Muse and Hollowsage.

On Arnold and Asia's untap step, Seedborn Muse sees that it is Asia's untap step and that she is another player so Arnold determines which permanents he controls untap as active player per the rules and again per Seedborn Muse's effect, then the turn-based action untaps them.

Do Arnold's permanents untap once or twice during that turn-based action?
If they do untap twice, does Hollowsage become untapped twice and trigger twice per 603.2c?