HCCI is basically combination of the 2. It’s compression ignition using gasoline instead of diesel. There is no spark to ignite the mixture. Instead, like a diesel, the density and temperature of the mixture are increased by compression until the mixture ignites spontaneously.
I thought that the consensus was that HAI/WAI works for some vehicles, and doesn't for others depending on engine and fuel system design. I don't think there's anyone here saying that it will work on any vehicle.