In our paper [here][1], Himes, Miller, Nariman, and myself prove that Hatcher-Vogtmann’s question has a negative answer, at least for $n=5$. It also probably has a negative answer for larger $n$, but our ignorance of the unstable cohomology of $Aut(F_n)$ prevents us from proving this via our techniques.  I do not know any reasonable conjecture as to what the dualizing module is.


  [1]: https://arxiv.org/abs/2203.03102