Equivalent locally finite form of Kuhn's unbounded strong realization conjecture

Let $M$ be an unstable module. For each integer $s$, let $\mathrm{R}_sM$ denote the reduced quotient associated with the $s$th stage of the nilpotent filtration, and call an unstable module of finite weight when its weight is finite. Say that $M$ is topologically realizable when it is isomorphic to the cohomology of a topological space.

Locally finite form of the unbounded strong realization conjecture. If $\mathrm{R}_sM$ is of finite weight for each $s$ and $M$ is topologically realizable, then $M$ is locally finite.

The source presents this as another form of the unbounded strong realization conjecture, using the preceding lemma relating membership in the Krull filtration to the modules $\mathrm{R}_sM$. No resolution is stated in the supplied text.