Stabilization-free extension theorem for controlled Mather–Thurston constructions

Let $(V,\partial V)$ be a manifold of dimension at least $3$. Let $X\to B\to V$ be a bundle with structure group $\operatorname{Homeo}(X)$, where $X$ is any manifold with a point-set metric. Suppose the bundle has, near $\partial V$, a topologically flat connection $\mathcal{F}_0$ whose holonomy lies in $I(X)$, the group of isometries of the fiber. Let $\mathcal{N}$ be any norm-topology neighborhood of $I(X)$ in $\operatorname{Homeo}(X)$. The stabilization-free extension conjecture. There exists a simply connected-sum cobordism $(W;V,V^\ast)$, constant near $\partial V$, covered by a bundle $\overline{B}$ with structure group $\operatorname{Homeo}_0(X)$ over a bundle $B^\ast\to V^\ast$, possessing a topologically flat connection inducing a representation

$$\rho:\pi_1(V^\ast)\longrightarrow\operatorname{Homeo}(X),$$

with $\rho(S)\subset\mathcal{N}$ for some generating set $S$ of $\pi_1(V^\ast)$. This should hold without stabilization. The claim is a proposed strengthening of the cited extension theorem: it would provide controlled holonomy after the construction without adding a stabilizing factor, but the supplied text gives no resolution or further evidence beyond presenting it as an optimistic conjecture.