Let $X$ be a finite dimensional $K(\pi,1)$ manifold. Recall that the space of unbased contractible loops $L_0(X)$ is the space of contractible continuous maps $S^{1}\to X$. There is an obvious $S^1$-action on $L_0(X)$.

Is it true that the quotient $L_0(X)/S^1$ is homotopic to $X$, and moreover the contraction can be realised as the contraction of $L_0(X)/S^1$ to the space of constant loops in it? What if $X$ is a finite dimensional $CW$-complex?

I think I can prove this statement in the case when $X$ is a smooth compact negatively curved manifold and we consider smooth maps from $S^1$ it. (I believe smoothness should not be essential.)

This is the follow up of The space of contractible loops of a finite dimensional $K(\pi,1)$