This is patently false. They will remove it if they are feeling particularly jolly when you call and ask. But they are under no obligation to do so and they have denied this action to me before.
If they refuse then you can just refuse to pay until they relent. Their only alternative is to take you to court or sell to the next collection agency down the line. Creditors and collection agencies won’t take that action except as an absolute last resort.
My personal experience has been that it is 100% true that, if paid in full, the item is removed.
Better?