After revisiting the heterotic string-derived low-energy effective model of cite{Ashfaque:2016ydg, Athanasopoulos:2014bba, Faraggi:2016xnm, Ashfaque:2016jha} constructed in the four-dimensional free fermionic formulation, we find two axions which are either harmful or massive. As a direct consequence, they can not solve the strong $CP$ problem which is in complete agreement with cite{Lopez:1990iq, Halyo:1993xn}. We also explore the possibility of the self-interacting dark matter residing in the non-Abelian gauge group present in the hidden sector cite{Faraggi:2000pv}. We find that the low-energy string-derived model naturally welcomes the self-interacting dark matter as $4$ copies of the non-Abelian, hidden $SU(2)$ gauge group factor are present.
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