Optimal complete-feedback card guessing under nonincreasing valley weights: A proof of the Diaconis-Fulman-Holmes shelf-shuffling conjecture

Let a random permutation $W\in S_n$ represent the order of $n$ distinct cards. After each guess, the player learns the actual card, and the objective is to maximize the expected total number of correct guesses. Diaconis, Fulman, and Holmes conjectured that, after a single uniform, unbiased shelf shu