An airfoil will achieve steady level flight if L = W and T = D. If its cruise speed is greater than orbital speed then it will lift itself beyond the atmosphere. But that's Mach 26 roughly for the altitude they have in mind. At those speeds the air is heated to 30,000 deg.K and moving by fast enough to transfer a big part of that heat to anything it touches. A gossamer structure would never survive.

***However***

I've also calculated a lift-to-drag ratio in excess of 50 at orbital velocity. This structure would definitely climb much higher at low - perhaps even subsonic - speed, resulting in zero to moderate heating only. At higher altitudes the heat transfer might be managable, even as speed becomes hypersonic. The question is, would it keep going up, up, up into space? Or would this L/D ratio decrease and at some point would the dynamic pressure tear it apart?

A serious analysis is required here. Atmospheric modelling, airfoil geometry & performance, and structural materials. A flight envelope and time-to-climb are the obvious algorithms needed.

I am withholding judgement on this concept. Without more info I can't say for certain if it would work or not. I for one would like to see it work. :-)