Mislin's finiteness-obstruction conjecture for finitely dominated H-spaces

Let $X$ be a finitely dominated H-space, meaning a pointed space with a map $X\times X\to X$ whose restriction to $X\vee X$ is homotopic to the folding map. Its finiteness obstruction is denoted by $\sigma(X)$. Mislin's conjecture. One has

$$\sigma(X)=0.$$

For $H$-spaces that are $\operatorname{CW}$-complexes, finite domination is characterized by finite generation of the direct sum of the integral homology groups, and the conjecture has been verified in a number of cases, including when the fundamental group is infinite or when the space has a classifying space.