The next tag line the party will push is women and domestic violence, as well as suicide. The point the government will be pushing is that women are victim of gun violence at home, and suicide by gun are happening because the gun is readily accessible at home.
They know that a ban on hunting rifles and shotguns will have very bad optics, but they feel they will be able to get away with central storage. The argument will be made that if the gun isn't readily available, it can save the lives of women and those who might re-think their suicide if they don't have their firearm handy, while not infringing on the rights of hunters by banning their firearms.