Semilocal simple connectivity of orbit spaces of smooth vector fields

Let $M$ be a Hausdorff manifold with a countable basis and empty boundary, and let $v$ be an arbitrary smooth vector field on $M$. Let $M/{\sim}$ denote the orbit space of $v$. Orbit-space semilocal simple-connectivity conjecture. The orbit space $M/{\sim}$ is semilocally simply-connected. This concerns the fundamental-group behavior of orbit spaces for arbitrary smooth vector fields, beyond the gradient-flow results established in the paper; the source does not provide evidence that the question has been resolved.