Simple connectivity conjecture for components of nondegenerate spherical-curve spaces

Let $\operatorname{CG}_{n+1}$ be the covering group used in the paper, and let ${\cal L}_n(z)$ be the corresponding space of nondegenerate spherical curves. A connected component means a path component of this space.

Simple connectivity conjecture. For every $z\in\operatorname{CG}_{n+1}$, every connected component of ${\cal L}_n(z)$ is simply connected.

The conjecture is motivated by the combinatorial description of the $1$- and $2$-skeleta of the relevant complex. The supplied text does not report a proof or disproof.