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In my paper, A variant of Mathias forcing that preserves $\mathsf{ACA}_0$ [Archive for Mathematical Logic 51 (2012), 751–780; arXiv:1110.6559, doi:10.1007/s00153-012-0297-4], I show the needed reals for $F_\sigma$-Mathias forcing are precisely the computable ones. In other words, for any non-computable set $C$ in the ground model there is a $F_\sigma$-Mathias generic $A$ that does not compute $C$. Of course, a $F_\sigma$-Mathias generic will be almost contained in or almost disjoint from any ground model set and hence all sets in $\mathcal{X}$.