Take a Hausdorff topological space $X$. Take two distinct points $x$ and $y$ of $X$. Consider a set $U$ of continuous paths $p$ from $[0,1]$ to $X$ equipped with the compact-open topology such that: $p(0)=x$, $p(1)=y$, $p$ is one-to-one and such that if $p\in U$, then $p\phi\in U$ for any nondecreasing homeomorphism $\phi:[0,1]\to [0,1]$ (let us call this group $G$).

Is the map $U\to U/G$ always a weak homotopy equivalence ? Or could someone provide a counterexample ?

I know cases where it is true. I am unsure that it is true in full generality and I'd be interested in seeing a counterexample.

Note: in fact, I work with $\Delta$-Hausdorff $\Delta$-generated spaces $X$ ($\Delta$-Hausdorff meaning that any continuous map $[0,1]\to X$ has a closed image), but I do not think that it is relevant here.