Look again at the first video (here,) and you will see that there are two discs, each one holding 4 pistons. The pistons are attached to the inner or outter disk at 90 degree intervals. About 30 degrees or so of the circumference of the piston wall is attached to the disk, making for what should be a very robust mounting.
However, the big problem I see long term for this engine is the same one that plagued the Wankel, which is lack of piston ring lubrication. In a conventional piston engine, the cylinder walls are coated with oil during the piston's upstroke, and the piston rings ride a film of oil on the downstroke.
In a Wankel, the three seals ride along the chamber walls dry, causing extreme wear issues. Same for this motor, no place to introduce cooling and lubricating oil on the cylinder walls, which will lead to longevity problems.
By disks, are you saying the two halves of the toroid are each one assembly, with four pistons each? Sort of two four-armed paddles, and that they oppose each others’ cumbustion pressures? To what do they react the combustion forces? Perhaps that is his cam arrangement opposite the toroid.
Regardless, there is no way that device is an internal combustion engine for very long. Imho.