Their primary argument is that regulation, including outright bans, of handguns "may reasonably be thought to preserve, not intrude on, ordered liberty." Of course, the Supreme Court ruled otherwise just last year. Never mind that—respondents are smitten with the phrase "ordered liberty," so much so that their brief uses it no less than 27 times.
Much of their work is concerned with contradicting the Heller decision, claiming that the right to keep and bear arms refers to a "right" of the states to arm militias, rather than the right of the individual to keep arms for self protection.
The scope of the Second Amendment right—weapons in common use—also reflects its purpose of protecting the militia, rather than an individual right related to self-defense, since the Second Amendment protects weapons regardless of whether they are useful for self-defense. p. 6
Yep, they're going to get real far with that.