Grodal's conjecture on the cohomology of simply connected polyGEMs

Let $X$ be a simply connected polyGEM of finite type at $2$, and let $\widetilde{\operatorname{H}}^*X$ denote its reduced cohomology with coefficients in $\mathbb{F}_2$. Grodal's conjecture. The reduced cohomology of $X$ is either trivial or not nilpotent. This conjecture proposes a strengthening of the known result that the reduced cohomology of a simply connected polyGEM of finite type at $2$, when nontrivial, is not locally finite as a module over the Steenrod algebra; the nilpotence question remains open in the supplied source.