There is a little problem (among others): meteorites. No material can survive such impact, not even CNTs. A tiny asteroid the size of a bean could easily cut the cable. And given the lenght of it such impacts would be rather frequent.
Also they should take a big carbon asteroid somewhere (i imagine from the asteroid belt) and bring it to Earh orbit. i see such task way beyond our current technological level.
Wonder as nobody pointed out such well known problems yet.