The work of Witsenhausen explores conditions under which two non-interactive users observing different coordinates of an i.i.d random process, can reach asymptotic agreement. Witsenhausen considers two scenarios: one in which both users observe a finite sequence with an error probability in the limit, and the second in which both users observe infinite-length sequences. In both cases, it turns out that perfect agreement is possible if and only if the joint distribution has a special decomposable structure known as a Gacs-Korner common part. This paper revisits Witsenhausen's work and makes two contributions. First, we show that both results are equivalent, that each implies the other. Second, we offer a new proof of the second result, that unlike the others, avoids any tensorizing arguments or manipulations of multi-letter information measures. We explain how this new converse might overcome some of the obstacles commonly encountered in the classical converse arguments.