Injured Player

If a player is injured and cut, does his injury time pass while available as a free agent or does it not start until he is picked up by a team? Can this be used as a strategy in high salary leagues where you maintain a player’s rights to your team only? What if you pick up another one of his cards to pass the time with? I have a player who is out for six games and I don’t want to experiment myself and risk losing the player for nine games.
Anyone’s experience here is appreciated.

A cut injured player's time will continue to pass one game at a time (3 games a night) while he's a free agent.

A cut injured player's time will continue to pass one game at a time (3 games a night) while he's a free agent.

Thank you!