Not any more, and planning around it is the most expensive mistake in polar travel. It used to work. Travellers would arrive in Ushuaia, walk the agency offices, and pick up an unsold cabin for a fraction of the list price. Agency offices there now see walk-ins almost daily and usually have nothing at all, or something needing a long wait. People fly south on the strength of a story they read, wait, and miss the season.
The reason it changed is dynamic pricing. Operators now watch their inventory and discount months ahead rather than dumping cabins at the last moment, because a pier full of unsold berths is a logistical problem they'd rather avoid. The discounting that used to happen three weeks out now happens 12 to 20 weeks out.
So the real last minute window is roughly three to six months before departure, not three weeks. That's when the price is generally as low as it goes.
The numbers people quote at us, a friend's 4,000 dollar trip or a 3,000 dollar figure from a video, are from an era that has ended. We'd rather tell you that plainly than let you gamble a season on it.