The score for this attraction has been arranged and adapted by Jonathan Sacks from Randy Newman's score for Monsters, Inc. Here is an article describing the scoring session which took place last autumn:
Thanks, Jim. Between the time I posted the question and seeing your answer, I stumbled across Paul Barrie's 'Window to the Magic.com' podcast show #26. This podcast features an audio flow through of the new ride and one can catch pieces of the score.