After having watched his movements in all of the scenes several times I'm thinking he is mostly flying but has a safety wire going up to a large high reach crane and, or possible helicopter for the lake flyover, that is just out of camera view that's working as a vertical stabilization assist.
The overall lack of any video shots from any distance makes me suspicious that there is something above him working as either a safety backup or a lift/stabilization assist of some sort.
To many of the movements just come off as being semi pendulum in action (left-right range and direction change plus pilot body lean angles Vs related platform tilt angles to make those movements just don't look untethered) as in being limited in range plus being stabilized by a crane and wire rather than full self-control.
Just watch the water powered Flyboard video and their movements as to how they direct the thrust to move and stabilize themselves. The air hoverboard guy's body movements are just too upright, stiff and slow on his reactions and counter movements to not have something holding him up and stabilizing him.
I say partial flying at best (thrust assisted hanging?) but with a definite hidden assist that's been edited out.
This may indeed be a hoax and they certainly didn't go out of their way to be convincing. No close-up of the device, no flashing of an explanatory slide with a cutaway drawing, no showing the fuel lines, no hovering over dry land, too much hokey music, and on and on.
But there are plenty of small jet engines around and using the control and stabilization scheme of a quadcopter and/or segue scooter would look a lot like that video.