The shell-type supernova remnant HESS J1731$-$347 emits TeV gamma-rays, and is a key object for the study of the cosmic ray acceleration potential of supernova remnants. We use 0.5-1 arcminute Mopra CO/CS(1-0) data in conjunction with HI data to calculate column densities towards the HESS J1731$-$347 region. We trace gas within at least four Galactic arms, typically tracing total (atomic+molecular) line-of-sight H column densities of 2-3$times$10$^{22}$ cm$^{-2}$. Assuming standard X-factor values and that most of the HI/CO emission seen towards HESS J1731$-$347 is on the near-side of the Galaxy, X-ray absorption column densities are consistent with HI+CO-derived column densities foreground to, but not beyond, the Scutum-Crux Galactic arm, suggesting a kinematic distance of $sim$3.2 kpc for HESS J1731$-$347. At this kinematic distance, we also find dense, infrared-dark gas traced by CS(1-0) emission coincident with the north of HESS J1731$-$347, the nearby HII region G353.43$-$0.37 and the nearby unidentified gamma-ray source HESS J1729$-$345. This dense gas lends weight to the idea that HESS J1729$-$345 and HESS J1731-347 are connected, perhaps via escaping cosmic-rays.
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