The global fixed-point conjecture for actions of homeomorphism groups

Let $M$ be a simply connected manifold that is not homeomorphic to $S^n$, and let $N$ be a manifold of dimension $n+k<2n$. The group $\operatorname{Homeo}_0(M)$ acts nontrivially on $N$. Global fixed-point conjecture. This action has no global fixed points. The conjecture is motivated by the observation that the local homology of the global fixed set contains the cohomology of $M$ with a degree shift; such large local homology appears difficult to realize. The claim is posed for general simply connected $M$ and its resolution is not known from the source.