Density conjecture for diffeomorphisms of the circle with non-abelian centralizer

Let $S^1$ be the circle, let $\operatorname{Diff}^1(S^1)$ denote the space of $C^1$ diffeomorphisms of $S^1$, and for $f\in\operatorname{Diff}^1(S^1)$ let $C(f)$ be its centralizer. Density conjecture. The set of diffeomorphisms $f\in\operatorname{Diff}^1(S^1)$ with a non-abelian centralizer is dense in $\operatorname{Diff}^1(S^1)$.