There are no exotic actions of diffeomorphism groups on 1-manifolds

Let $M$ be a manifold, $N$ 1-dimensional manifold. Assuming $r \neq \dim(M)+1$, we show that any nontrivial homomorphism $\rho: \text{Diff}^r_c(M)\to \text{Homeo}(N)$ has standard form: necessarily is $1$-dimensional, and there are countably many embeddings $\phi_i: M\to N$ with disjoint images such the action of $\rho$ conjugate (via product $\phi_i$) to diagonal $\text{Diff}^r_c(M)$ on $M \times M ...$ $\bigcup_i \phi_i(M)$, trivial elsewhere. This solves conjecture Matsumoto. We also groups have no countable index subgroups.