Whittaker's conjecture on morphisms of homeomorphism groups

Let $M$ be a compact manifold, and let $\mathrm{Homeo}_{0}(M)$ denote the identity component of its homeomorphism group. A group morphism is trivial if it maps every element to the identity; a morphism is induced by conjugacy by a homeomorphism $h:M\to M$ if it maps $f$ to $h\circ f\circ h^{-1}$. Whittaker's conjecture. Every group morphism

$$\mathrm{Homeo}_{0}(M)\longrightarrow\mathrm{Homeo}_{0}(M)$$

is either trivial or induced by conjugacy by a homeomorphism. This extends the rigidity phenomenon for group isomorphisms established by Whittaker. The conjecture is solved for the circle, while the analogous statement for general compact manifolds remains open.