Constant expected search complexity for CCL-coded fast forward permutations

Let $(a_0,a_1,\dots,a_l)$ be a sequence generated by the CCL procedure, and let $\pi$ be the fast forward permutation coded by this sequence. Choose $x\in\{0,\dots,N-1\}$ uniformly. The naive algorithm compares $x$ successively with $s_0,s_1,\dots$ until it finds the first $i$ such that $x<s_i$. Constant-complexity conjecture. The expected value of this index $i$ is $2$. Consequently, the complexity of computing $\pi^m(x)$ is $O(1)$. This claim concerns the average-case performance of the naive search algorithm for the cycle data produced by the CCL procedure, in contrast with the preceding worst-case logarithmic bounds.