The stronger hemispherical-curve conjecture for locally convex curves on the 3-sphere

Let $\mathcal{L}\mathbb S^3(\mathbf{1},-\mathbf{1})$ be the space of locally convex curves on the 3-sphere with the indicated endpoint data, let $L$ denote the associated map to curves, and let a borderline hemispherical curve have rotation number defined as in the paper. The stronger hemispherical-curve conjecture. The image of the whole space $\mathcal{L}\mathbb S^3(\mathbf{1},-\mathbf{1})$ by $L$ does not contain a borderline hemispherical curve with rotation number equal to $2$. This would make the necessary condition for a curve in $\mathcal{L}\mathbb S^3(\mathbf{1},-\mathbf{1})$ to be convex sufficient; the authors state that they are not currently able to prove the stronger statement.