Not being of the legal persuasion, we have no idea if the judge will allow this, as it would set a fairly dangerous precedent for all auto makers. Imagine if every time a car got recalled for a faulty cup holder (or whatever) the owners didn't have to pay for it. While it might sound like music to a recalled person's ears, corporations have a nasty habit of massing them sorts of costs back onto the consumers en masse, meaning we all could suffer."Until they make it safe, people shouldn't have to pay their lease payments and shouldn't have to make their bank payments. Let Toyota pick that up."
Though perhaps this proposed class action suit (the second one concerning ToMoCo we're aware of) will be contained just to Toyota. Here's a bit more of what the suit alleges, "[I]t was more important to Toyota to increase sales and become the largest manufacturer in the world. Toyota sacrificed innocent, trusting lives for profit and hubris." Your Volvo's stall-prone software is hardly in the same league.
The suit doesn't stop there. Another cited reason for Toyota to pick up the monthly payment tab is because all this recall hysteria has hurt the value of the cars in question. Frankly, and long term, that might be the most interesting point of all. The suit also claims that Toyota has previously "destroyed potentially incriminating documents" and wants the court put a stop to the alleged practice.