In <a href="https://mathoverflow.net/questions/77836/gods-number-for-the-n-times-n-times-n-cube">this</a> MO question, user Martin Brandenburg asks about God's number for $n \times n \times n$-cubes for $n>3$. Here, God's number $g(n)$ was defined as the smallest number $m$ such that every realizable arrangement of the $n \times n \times n$-cube can be solved within $m$ moves. 

We could <a href="https://en.wikipedia.org/wiki/N-dimensional_sequential_move_puzzle">generalize</a> the notion of God's number to $n^{k}$-dimensional cubes, where $k >3$. Define $g_{k}(n)$ as the smallest number $l$ such that every realizable arrangement of the $n^{k}$-cube can be solved within $l$ moves.

Whereas Mr. Brandenburg's question pertains to three dimensional Rubik's cubes ($k=3$), I wonder what is known about higher God's number for higher dimensional sequential move puzzles. 

**Questions**: 

 1. Is $g_{4}(2)$ known? And what about $g_{4}(3)$? 
 2. What is known about the asymptotic value of $g_{k}(n)$?