Noncentral component conjecture for nondegenerate spherical curves

Let $\operatorname{CG}_{n+1}$ be the covering group used in the paper, with center $Z(\operatorname{CG}_{n+1})$, and let ${\cal L}_n(q)$ denote the space of nondegenerate spherical curves with parameter $q$. The group $\operatorname{Spin}_{n+1}$ has based loop space $\Omega\operatorname{Spin}_{n+1}$.

Noncentral component conjecture. If

$$q \in \operatorname{CG}_{n+1}\smallsetminus Z(\operatorname{CG}_{n+1}),$$

then the inclusion

$$i_q:{\cal L}_n(q) \to \Omega\operatorname{Spin}_{n+1}$$

is a weak homotopy equivalence.

The conjecture predicts that all noncentral components have the homotopy type of the based loop space, leaving at most two or four other homotopy types according to the parity of $n$. The text cites the classification for $n=2$ as supporting evidence, but gives no general resolution.