Converse characterization of abelian duality spaces via the Albanese map

Let $X$ be an $n$-dimensional smooth complex quasi-projective variety with proper Albanese map

$$\operatorname{alb}:X\to\operatorname{Alb}(X).$$

An abelian duality space of dimension $n$ is a space whose cohomology with coefficients in the group ring of its first homology is concentrated in degree $n$ and is nonzero there. Converse conjecture. $X$ is an abelian duality space of dimension $n$ if and only if $\operatorname{alb}$ is semi-small and $X$ is homotopy equivalent to a finite $n$-dimensional CW complex. The theorem preceding this conjecture proves that the abelian-duality property follows under the stated geometric conditions in one direction; the conjecture asserts the converse as well, but the source gives no resolution.