Finite-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 $H^*X$. Finite-profile conjecture. If $q_X$ takes finite values, then $H^*X\in\mathscr U_0$. The source identifies this as a stronger conjecture equivalent to the unbounded strong realization conjecture of Kuhn; no resolution is stated here.