I think I see how the elevon controls are designed. The pulleys by the pilot's waist look independent, except for being connected to the sides of the "pillow" the pilot rests his torso on. So that creates linkage between controls for side-to-side or forward-backward movement.
The pilot moving his body (and thus the pillow) left or right will pull on one and create slack in the other, creating opposite changes in elevon position. This gives roll control.
Near the end of the video, after landing (e.g. at 5:00) we see a good angle of the "pillow" positioned so the cord attachment points are as close as they can get to the pulleys. So the pull on each cord is at a minimum. This is presumably downward deflection of both elevons since the pilot moved to this position after touchdown. Moving left or right could probably lower one a bit further (while raising the other). (In that same shot, we can see the elevons in frame as well, and it looks like they're not far down, but this is a bad angle. Maybe they simply don't go very far down, or there's some kind of internal linkage, or I'm missing some other control input.)
But in flight, I think the pilot generally keeps his body position farther back, so both ropes are pulled out some. This pulls the elevons up to a neutral or higher position, leaving room for pitch up or down by moving the pillow back or forward, respectively. (Along with the pilot's weight, which also provides some pitch control directly, apart from moving the elevons).
In this position, moving left and right will still slacken one and tighten the other, creating the opposite movements you want for roll control.
During landing (4:22), we have a camera angle from above/behind the pilot that shows both ropes and pulleys while he's moving, controlling pitch while maintaining some right-roll. Note how he pushes backward (tensioning the ropes) to flare a bit just before touchdown.
To make roll control work in the correct orientation, I think the right rope must be connected to the left elevon, and vice versa. Because moving your body (including bodyweight) right slackens the right rope. But to roll right, you need to deflect the right elevon up. If it was right rope => right elevon, either roll or pitch control would have to work against the pilot's weight shifts, depending on whether pulling deflected upward or downward. That would be terrible and unintuitive, and we can see from the video that he's moving and/or leaning right to roll right, and back to pitch up.
This is my explanation of how I think it's designed, not based on any statements from the designer. (But thanks to the other answers for their observations that shaped my thinking, especially @quietflyer for spotting the pulleys.) I think it's a plausible design for the controls, compatible with sanity and all the observations I could make from the video, but possibly not the actual design.
Certainly body-weight shifts are part of the controls; as @quietflyer says we can't be sure how much of the control authority comes from body-weight vs. elevons. But it looks like there is good control of the elevons for both pitch and roll.
2:50 in the video is a decent side view of exercising some pitch control (near the ground so it provides a good reference). We definitely see the pilot shift his body forward and back.
It's hard to see the far elevon. I can't tell whether it's moving as much (in the opposite direction) as the left aileron goes up and then down. If not, that would prove that these are elevons which can also do pitch control, not just linked ailerons. I think that's the case from what would make sense for the rope/pulley design, but I'm not 100% sure.