Recall that the covering number $cov(B)$ is the least cardinal $\kappa$ such that $\kappa$ meagre sets cover the real line. Andreas Blass and John Irwin [http://www.math.lsa.umich.edu/~ablass/bb.pdf][1] prove, among many other results, that $\mathbb{Z}^{\omega}$ is never the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$, if the chain has length less than $cov(B)$. They note it might be possible to remove the hypothesis on the length of the chain. However, they raise the **question** whether e.g. under $CH$, $\mathbb{Z}^{\omega}$ could be the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$. If the chain $\lbrace A_{\alpha} : \alpha < \delta \rbrace$ has the additional property that $\mathbb{Z}^{\omega}/A_{\alpha}$ is cotorsion-free for all $\alpha$, the answer is negative. 

Has there been any further progress on this question?


  [1]: http://www.math.lsa.umich.edu/~ablass/bb.pdf