[infogulch]

After a bit of research I found that there is a proof [0][1] that BB(12) > g_1. But that's still a ways from g_2, let alone g_64.

[0]: "A Lower Bound on Rado's Sigma Function for Binary Turing Machines" by Milton Green (1964)

[1]: https://en.wikipedia.org/wiki/Busy_beaver#Known_values_for_....