Yes, the ring flip is the key. Good work.
As fun as pericyclic reactions are, this was a conformational analysis problem at heart. The oxy cope needed the six-membered chairlike transition state to get the desired stereochem, and the ring flip was important for the ene.
I'm not sure why the shown diastereomer is the product of this exercise. Perhaps just to get people to think conformational-ly. In the event, the diastereomer that Dan drew (and I drew when I first attempted the mechanism) is the favored mechanism in the reaction. In the paper, dr's range from 2.2:1 (alcohol up) to > 25:1 (alcohol up). http://dx.doi.org/10.1021/ja066830f
In any event, since we have time left this week, here's a follow up questionFollow Up QUESTION: In a non-tandem process, the oxy-cope is known for similar compounds with just the free alcohol (that is, without the allyl group - see figure). If the rate for the free-alcohol version of the oxy-cope is set at K1 = 1, the anionic oxy-cope benefits from a significant rate enhancement, K2 = 1010-1017.