Rectangular inscription conjecture for spherical Jordan curves of diameter π

Let $S^2$ be the unit round sphere, and let $\gamma$ be a smooth Jordan curve in $S^2$. Its diameter is the supremum of the round distances between points of $\gamma$; assume that this diameter is $\pi$. A rectangle of type $\theta$ is the inscribed configuration defined in the paper with parameter $\theta\in(0,\pi)$. Spherical rectangular inscription conjecture. The curve $\gamma$ inscribes a rectangle of type $\theta$ for every $\theta\in(0,\pi)$. This extends the preceding theorem beyond curves disjoint from their antipodals, and would establish rectangular inscriptions for smooth spherical Jordan curves whose diameter is $\pi$; the proposed approaches involve Hamiltonian motion and Floer-homological ideas.