The Naive Conjecture on approximate fibering modulo tight torsion

Let $p\colon M\to B$ be a continuous map satisfying the standing assumptions: $M$ is a connected closed topological manifold; $B$ is an aspherical closed topological manifold admitting a PL structure; the homotopy fiber of $p$ has the homotopy type of a finite CW complex; and $p$ induces a surjection on fundamental groups. Let $N\tau(p)$ denote the tight torsion in the cokernel of the relevant assembly map. Naive Conjecture. The map $p$ is homotopic to a MAF if and only if $N\tau(p)=0$. The conjecture proposes that vanishing of tight torsion is the only obstruction to approximate fibering. It is false in general, as the paper states that the Naive Conjecture is wrong in general; stronger hypotheses, such as the fibered conjecture under Farrell–Jones and orientability assumptions, yield positive results.