Bounded profile conjecture for nilpotent cohomology

Let $X$ be a space such that the reduced cohomology $\tilde H^*X$ is nilpotent. Let $q_X=w_{QH^*X}$ be the profile function of the unstable module of indecomposable elements of $H^*X$. Bounded profile conjecture. If $q_X$ is bounded, then $H^*X otin\mathscr U_0$. This conjecture is presented as evidence-based on the equivalence theorem relating profile functions and membership in $\mathscr U_0$; its resolution is not specified in the source.