The non-equivalence conjecture for spaces of locally convex curves

Let $n>3$ and let $q otin Z(\operatorname{Quat}_{n+1})$. Consider the space ${\cal L}_n(1;q)$ of locally convex curves with endpoint data $(1,q)$. If it contains convex curves, let ${\cal L}_{n,\operatorname{non-convex}}(1;q)\subset {\cal L}_n(1;q)$ denote the connected component of non-convex curves. Non-equivalence conjecture. The space ${\cal L}_n(1;q)$ is not homotopically equivalent to $\Omega\operatorname{Spin}_{n+1}$. Moreover, if there are convex curves in ${\cal L}_n(1;q)$, then ${\cal L}_{n,\operatorname{non-convex}}(1;q)$ is also not homotopically equivalent to $\Omega\operatorname{Spin}_{n+1}$. This is proposed as a claim for dimensions greater than three, contrasting with the preceding results and with the stated corollary; the source does not establish it.