Unless I'm missing something, five field kono should have been cooked long ago.
Because only diagonal moves are allowed, the board consists of two independent fields:

"odd" field (13 squares) with 3 R pieces, 3 B pieces, and 7 empty squares, so
13! / 3! 3! 7! = 34230 possible states

"even" field (12 squares) with 4 R pieces, 4 B pieces, and 4 empty squares, so
12! / 4! 4! 4! = 34650 possible states

The full game is the composite of the two fields.  Even if all possible combinations
had to be analyzed, that's only 1,189,188,000.  But each field can be analyzed separately,
and then an analysis of the full game can be made much more simply than by examining
all possible combinations, no?  (Hand waving goes here... :-)

-- Mike

ps. there is a nice five field kono player at  http://www.darkfish.com/kono/FiveFieldKono.html