Connectedness conjecture for real spherical tight-frame spaces

Let $\mathcal{F}^{\mathbf R}_{k,n}$ denote the space of real spherical tight frames with $k$ frame vectors in dimension $n$. Connectedness conjecture. For $k,n\in\mathbf{N}$ with $n\ge2$ and $k\ge n+2$, the space

$$\mathcal{F}^{\mathbf R}_{k,n}$$

is connected. This is suggested by the connectedness results for the cases established earlier in the paper; the conjecture concerns the remaining general real case.