Kuhn's dichotomy conjecture for the functors associated to unstable cohomology

Let $X$ be a space. For each $s$, let $R_s(H^*X)$ denote the analytic functor associated with the reduced unstable module $R_s(H^*X)$. Kuhn's dichotomy conjecture. Either at least one of the functors $R_s(H^*X)$ has an infinite Loewy (socle) series whose composition factors are unbounded in degree, or all the functors $R_s(H^*X)$ are constant. In terms of unstable modules, this would imply that at least one corresponding unstable module has infinitely many generators. The conjecture is stated as a reformulation of the most general form of Kuhn's conjecture, and the paper gives no resolution.