God's number for the $n \times n \times n$-cube
Let $g(n)$ be the smallest number $m$, such that every realizable arrangement of the $n \times n \times n$ cube can be solved with $m$ moves. In other words, this is the "radius" of the Cayley graph of the $n \times n \times n$ cube group with respect to the canonical generating system.
We have $g(1)=0$, $g(2)=11$ and - quite recently - in 2010 it was proven that $g(3)=20$: God's number is 20.
Question. Is anything known about $g(4)$ or $g(5)$?
I expect that the precise number is unkwown since the calculation for Rubik's cube already took three decades. Nevertheless, is there any work in progress? Are any lower or upper bounds known?
I would like to ask the same question about $g(n)$ for $n>5$, or rather:
Question. Is anything known about the asymptotic value of $g(n)$?