Well, since you insist, here are my lower bounds for the 4^3. For
the "colored" 4^3, where only the color pattern matters, some
positions require at least 41 qtw to solve. For the "marked" 4^3,
where center facets of the same color are distinguished so as to
force a unique home position for each, some positions require at
least 48 qtw. The proof is in MC:ALAN;CUBE4 LB and is about 5K
characters long.

A qtw of the 4^3 is either a quarter-twist of a face relative to
the rest, or a quarter-twist of half of the puzzle relative to the
other half. Note that this makes a slice twist into two moves. I
like this metric because it is consistent with our conventions for
the 3^3. One of these days I'll explain why I like those
conventions.