The room temperature resistance of an individual SWNT (single wall carbon nanotube, armchair) can be as low as 1e4 Ω/μm or 1e10 Ω/m . The minimum diameter SWNT (3,3) is about 0.43 nanometers; about 50 carbon atoms per nanometer, or 5e10*2e-26 kg/m or 1e-15 kg/m per SWNT. A 1kg/m bundle of SWNT (3,3) will have a resistance of 10 μΩ/m. The resistivity of aluminum is 2.65e-8 Ω/m and the density is 2700 kg/m³, so the resistance of a 1 kg/m wire is 7 μΩ/m. Iodine-doped MWNT (multiple wall nanotubes have reduced resistance, but annealing bakes out the iodine; if heating permanently increases resistance, (which leads to more heating), this is not a safe material for high current conductors.
Optimistically, someday we may have large perfectly aligned CNT power cables with stable resistivities lower than aluminum, but they won't be much lower, due to quantum limits and thermal scattering of ballistic electrons.
I need to find trustworthy references (I've found many with bonehead math mistakes) to verify the above, but in 2017, no lab results for single nanotubes exceed the conductivity of bulk aluminum, which is expensive, but vastly less expensive per kilogram than aligned carbon nanotubes. Do not count on miraculous improvements in CNT resistivity salvaging an inadequate engineering design.