Thank you for the article! The chain-walking (or chaining in general) assumes that the permutation is full-period. For LCGs, the properties you specified indeed guarantee that (by Hull–Dobell Theorem). I wonder, though, whether this is true for the Kensler permutation as well. I couldn't find any reference to it in the linked article or the original paper. Cheers!
The cycle-walking of the Kensler permutation does not require a full period, so that's probably why you couldn't find this information. :-) In fact, the permutations it's typically built on, including mine, have lots of independent cycles, and even fixed points. How could this possibly work? It's subtle, and it's tripped me up every time I revisit it: The starting index is guaranteed to be in range since the caller must only ask for an index in range. Before considering this index, it's permuted, potentially going out of range. Eventually this cycle-walk must come back into range since it started in range with the original index.