The Busemann biplane came up when we did supersonic aero in University back in '98 or '99 and it was always stated to be an impractical wing design because, at the supersonic zero boom/zero wave drag condition, it couldn't produce lift; this doesn't stop it being useful for other things like shells etc. where you don't mind zero supersonic lift if you can get low drag
The diagrams in the article seem to look like that condition in supersonic flow where the "inner" surfaces interfere favourably with each other to cancel wave drag and have the upper and lower surfaces with no incidence to the flow so they produce no shock waves.
Supersonically it should still produce lift quite happily if you angle it so there is incidence to the flow but I think that it should then produce wave drag and booms... For example I can't see from the article how, in a lifting condition, the shock wave from the compression of the supersonic flow on the undersurface (which produces the compression & higher pressure that helps lift the wing) could be cancelled out without having another wing underneath that; then you have the same problem with the undersurface of that wing & then you're in a "it's wings all the way down" problem.
Conventional 'low boom' solutions (like the Gulfstream/NASA "quiet spike") all tend to shape the nose of the aircraft to reduce the suddenness of the pressure increase across the shock wave but they aren't able to eliminate it...
It could be that they've found a case where they can get low wave drag/boom while still producing some lift and also getting decent subsonic lift/drag - that would be really interesting...