Let $X$ be a finite dimensional $K(\pi,1)$ manifold. Recall that the space of unbased null-homotopic 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 homotopy equivalent to $X$? Moreover, is there is a deformation retraction from $L_0(X)/S^1$ 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 https://mathoverflow.net/questions/296670/the-space-of-contractible-loops-of-a-finite-dimensional-k-pi-1