I never did take the geosync rope seriously. Hence Paul Birch and his orbital ring, with Kevlar (or Zylon). Tonight I've been alerted to the selenosync rope. Unlike the cables to the Galapagos or Nairobi lifts, Elon promises to talk with us about Sinus Medii's cable.
Luna doesn't exactly have a "LEO" like our GEO but she does share with us a (t)LL1. (And L2, darkside.) Also see Jeremy Hsu from 2019.
First problem: 'tis halo. One could stationkeep it with aluminum/oxygen thrusters, but that seems likely to dust up the region. "Gamma Factor" notes a better idea: that this Luna-focused rope will need a counterweight, as we're pulling toward the Moon whenever we use it. That's another rope dangling toward Earth - the Spaceline.
GEO is uninhabited by space-stations today. Only unmanned sats float there. So shall GEO ever be.
The spaceline does not dip into our GEO unless GEO hosts an orbital ring, which it probably won't ever. The idea is that the Earthside point of the counterbalancing tether get its own little station, mostly just a hook for the (long) pull up to L1. From GEO to this hook requires a little boost through vacuum. Which shouldn't be that bad compared to getting from Earth to GEO in the first place. In fact small cargo might be possible by railgun or spinlaunch rather than rocket.
As to the cable, the Shenzhen trio Wang Weiweia, Wu Zhiganga, and Liu Jiafu have a paper "Conceptual Design and Mechanical Analysis of a Lunar Anchored Cislunar Tether". This assumes an internal tension of about two kilonewtons along the L1 plateau: Kevlar or some other 2.5+% elastic. Kevlar counterweight dangles as far as 110 Mm over the Earth so 114.8 Mm semimajor. Under-elastic T1100G could, I think, bring us down to the 80s Mm (extended to its maximum 2.2%); but there's still tension for the Lunar side of L1 pivot. So here we might have to swap it out for, I dunno, Dyneema somewhere on the Lunaside where the tension be less. Of course then someone has to pay for all that cable.
UPDATE 6/17/26 David Jensen keeps citing a mammoth powerpoint from Bell and Hines (2012 ppt) that anhydrous glass can reach 13.8 GPa - that's a giga. I couldn't find this number in the document which anyway is a... powerpoint, not a study. However I do find studies from 1993 which point to 13+ GPa at 34.5μm width (pdf). Jensen honestly should point to those. Anyway a fiberglass that is four times as strong will yield 5700 N breaking pull for 1 g/m3, although anhydrous might be denser. But we are assuredly going down to the 60s Mm even 50s Mm.
REWRITE 7/20/23 - ugh this poast "GEO space-stations won't happen" was baaad. I've redone it. AND AGAIN 6/17/26 because there's more, tucked into the appendix.
ABOUT L2 5/18/24 - This poast left alone what to do with the far side of the Moon and, three years later, I'm still stumped. Its tangential delta-V against the ecliptic is too powerful for anything out to Vesta. It can lob parts of a longterm-transport at (say) Ceres, to be assembled en-route. So... maybe that?