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The tunneling reaction H$_2$ + D$^-$ $\rightarrow$ HD + H$^-$ was studied in a recent experimental work at low temperatures (10, 19, and 23~K) by Endres {\it et al.}, Phys. Rev. A {\bf 95}, 022706 (2017). An upper limit of the rate coefficient was found to be about 10$^{-18}$ cm$^3$/s. In the present study, reaction probabilities are determined using the ABC program developed by Skouteris {\it et al.}, Comput. Phys. Commun. {\bf 133}, 128 (2000). The probabilities for ortho-H$_2$ and para-H$_2$ in their ground rovibrational states are obtained numerically at collision energies above 50~meV with the total angular momentum $J$ = 0 - 15 and extrapolated below 50~meV using a WKB approach. Thermally averaged rate coefficients for ortho- and para-H$_2$ are obtained; the largest one, for ortho-H$_2$ is about $3.1\times10^{-20}$ cm$^3$/s, which agrees with the experimental results.