fwiw I did most of the work checking the mystery team's 67 RPI (I'll refer to them as TEAM). I'm not going to calculate adjusted winning % for all the opponents and opp' opp, but I got extremely close to the correct RPI without doing that. It's definitely working as advertised imo.
I think the main reason TEAM's RPI seems better than you might expect is the large number of road games, which count for slightly more in RPI. So they're 11-10, but when you apply the weights for home/road games from the FAQ I posted, their "adjusted" record is 13.8 wins and 9.2 losses. Again that is not reflecting anything about their conference, just the fact that they've played 15 road games and only 6 home games. So 13.8 - 9.2 is a 60% winning%, which is 25% of the RPI formula.
50% of the RPI formula is opp winning%, which I have as .5913 for TEAM. That's mainly due to playing in a BCS conf and having some extremely good W-L records on their schedule. I did not adjust the opp winning% for home/road games, but it would make only a small difference.
The last 25% of the formula is opp' opp winning%, and I have TEAM at .5249 there. Again I did not adjust for home/road and it would make a small difference.
I get .5769 as TEAM's RPI. The actual listed value is .5810. Most of their BCS conf opponents probably played lots of road games too, so I'm confident that is all the difference is. I'm slightly understating their RPI by not adjusting for home/road on their opp and opp' opp.