There is no material with the tensile strength to be able to handle this.

It would just topple over anyway, wind isn’t forgiving, you can’t have a solid foundation halfway up the sky.

Every action has an equal and opposite reaction.

Objects in motion stay in motion until acted upon by an outside force.

Given the above two rules, the elevator would rip the structure into shreds. We default back to the tensile strength part again. There is no material capable of holding itself up that high, as well as withstanding the force moment exerted by a high speed elevator.

Maybe instead of reading from a site called ScientificAmerican, consult an engineer.