Let $M$ be a connected (topological) oriented $m$-manifold (say without boundary), and let $\operatorname{Homeo}^+(M)$ be the group of orientation preserving homeomorphisms $M \to M$.

Is it true that for every $f \in \operatorname{Homeo}^+(M)$, there is a locally flat oriented embedding $\Phi \colon \mathbb{D}^m \hookrightarrow M$ and $F \in \operatorname{Homeo}^+(M)$ such that $F$ is isotopic to $f$ and $F \Phi = \Phi$?