Existence of paths detecting every element of the continuous-contingent microbundle space

Let $\mathsf{W}_{\mathrm{CC}}$ be the space of continuous-contingent microbundles, let $u\in\mathsf{W}_{\mathrm{CC}}$, and let $\mathrm{D}(u)$ denote its domain. A path $\omega$ detecting $u$ is a path for which the evaluation map $\mathrm{ev}(\omega(t))$ is defined as a set-valued function on $\mathrm{D}(u)$. Detection-path conjecture. For every $u\in\mathsf{W}_{\mathrm{CC}}$, there is a path $\omega$ detecting $u$ such that

$$\left(\mathrm{ev}\left(\omega(t)\right)\right)(s)\neq\varnothing\qquad\forall t\in(-1,1)\setminus\{0\},\quad\forall s\in\mathrm{D}(u).$$

The conjecture asserts the existence of a detecting path whose evaluations are nonempty at every nonzero parameter and every point of the domain. Its status is not determined by the supplied source context.