First, to answer the question: A clock on the Moon is faster by
57 microseconds per Earth day. (See
https://www.newscientist.com/articl...on-the-moon-by-57-microseconds-per-earth-day/ ) So, that is a ratio of 1.00000000066 Moon Time to Earth Time.
It seems odd to me if scientists would propose a "Moon Time" that has a constantly changing relationship to our standard "Earth Time" which we call by various names such as "Greenwich Mean Time", "Coordinated Universal Time", etc.
Why not just recognize that we all need to deal with various reasons that our local times deviates from that "universal" standard in well defined ways.
So, we could just make clocks on the Moon run faster by 0.000000066%. For astronauts looking at their watches, they would just need to turn them back by one second about every 50 years to stay in sync with Earth-based UCT. That shouldn't be much of an issue in an environment where sundowns and sunrises are space about 14 Earth days apart. Any lunar base would still need a lunar site specific table of rise and set times, anyway.
For actual navigation, UCT with the necessary relativistic corrections, which we already do for GPS satellite info on Earth, would work fine, instead of "mission time", for coordination among various missions, just like we can do with aircraft today. But, from the standpoint of things like consumption of expendables on missions, "mission times" are still probably useful. There would just be a well understood relationship between mission times and UTC.