Conjecture on actions of circle homeomorphisms on spheres and discs

Let $\mathbb{S}^{1}$ be the circle, $\mathbb{S}^{2}$ the sphere, and $\mathbb{D}^{2}$ the closed disc. Consider non-trivial actions of the group $\mathrm{Homeo}_{0}(\mathbb{S}^{1})$ on these surfaces, and let $\varphi_{K,\lambda}^{\mathbb{S}^{2}}$ and $\varphi_{K,\lambda}^{\mathbb{D}^{2}}$ denote the families of actions referenced in the source. Sphere-and-disc action conjecture. Any non-trivial action of $\mathrm{Homeo}_{0}(\mathbb{S}^{1})$ on the sphere is conjugate to one of the actions $\varphi_{K,\lambda}^{\mathbb{S}^{2}}$, and any non-trivial action on the closed disc is conjugate to one of the actions $\varphi_{K,\lambda}^{\mathbb{D}^{2}}$. The conjecture is motivated by the classification theorem for actions on the closed annulus and torus, but the source gives no resolution for the sphere or closed-disc cases.