Professor says we could already build a floating space elevator

If this is of interest, you might check out Science Futurism with Isaac Arthur.

His Upward Bound series - examining methods for getting into space faster, cheaper, and safer:

https://www.youtube.com/playlist?list=PLIIOUpOge0LsGJI_vni4x...

and related sites:

https://isaacarthur.net/

https://www.youtube.com/channel/UCZFipeZtQM5CKUjx6grh54g

https://www.listennotes.com/podcasts/science-futurism-with-i...

One problem I see with this approach is that the two endpoints will have different heights, and therefor different orbital periods[0]. This will put a lot of strain on the cable between them. This puts a limit on the length of the elevator.

[0] https://en.wikipedia.org/wiki/Geocentric_orbit#Tangential_ve...