Alternate Bing–Borsuk conjecture via approximate fibrations

Let $X$ be nicely embedded in $\mathbb R^{m+n}$ for some $m\geq 3$, and suppose that $X$ has a mapping cylinder neighborhood $N=C_\phi$ of a map $\phi:\partial N\to X$, with mapping cylinder projection $\pi:N\to X$. An approximate fibration is a map having the approximate homotopy lifting property. Alternate Bing–Borsuk conjecture. The projection $\pi:N\to X$ is an approximate fibration. This formulation is stated as equivalent to the Bing–Borsuk conjecture and remains open with it.