Freedman's controlled Mather–Thurston conjecture

Let $(V,\partial V)$ be a manifold with $\dim(V)\geq 3$. Let $X\hookrightarrow B\to V$ be a bundle with structure group $\operatorname{Homeo}(X)$, where $X$ is a manifold with a point-set metric, and suppose the bundle has near $\partial V$ a topologically flat connection $\mathcal{F}_0$ whose holonomy lies in $\operatorname{Iso}(X)$. Let $N$ be any norm-topology neighborhood of $\operatorname{Iso}(X)$ in $\operatorname{Homeo}(X)$. Freedman's controlled Mather–Thurston conjecture. There exists a one-sided $s$-cobordism $(W,V,V^{\ast})$, constant near $\partial V$, covered by a bundle $\overline{B}$ with structure group $\operatorname{Homeo}_{0}(X)$ to a bundle $X\hookrightarrow B^{\ast}\to V^{\ast}$ possessing a topologically flat connection inducing a representation $\rho:\pi_1(V^{\ast})\to\operatorname{Homeo}(X)$ such that $\rho(S)\subset N$ for some generating set $S$ for $\pi_1(V^{\ast})$. The conjecture is proposed as an extension of Freedman's result; the source does not establish it in the stated generality.