Degree-growth conjecture for the functors associated to unstable cohomology

Let $X$ be a space, and let $d$ be the least integer such that $R_d$ is a nonconstant functor of finite degree. Assume that $d>1$. Degree-growth conjecture. At least one of the functors $R_s$, with $d<s<2d+1$, has unbounded degree. This is presented as a refinement of the preceding reformulation of Kuhn's conjecture. The paper supplies no proof or resolution of this refinement.