Diagonal Poincaré embedding conjecture for finitely dominated Poincaré spaces

Let $M$ be a connected Poincaré duality space of dimension $d$ satisfying Hypothesis: either $M$ is homotopy finite, or $M$ is finitely dominated and the Bass trace conjecture holds for $\pi=\pi_1(M)$. Assume $d\geq 4$ and let $\Delta\colon M\to M^{\times 2}$ be the diagonal map. Diagonal embedding conjecture. The map $\Delta$ admits a Poincaré embedding. The preceding corollaries establish this in several cases, including certain abelian fundamental groups and groups whose nontrivial elements all have odd order; the conjecture seeks the same conclusion under the stated general hypotheses.