ANR homology 4-manifold conjecture for Poincaré 4-complexes with free fundamental group

Let $X^4$ be a Poincaré $4$--complex whose fundamental group is a free product of $p$ copies of $\mathbb{Z}$:

$$\pi_1(X^4)\cong \mathop{*}\limits_1^p \mathbb{Z}.$$

Free-fundamental-group conjecture. Then $X^4$ is (simple) homotopy equivalent to an ANR homology $4$--manifold.

The preceding discussion establishes that the total surgery obstruction vanishes for the Poincaré 4-complexes considered there, making this a proposed special case of the broader four-dimensional surgery problem. The supplied source gives no resolution.