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We study a population of significantly sub-solar enrichment galaxies at z  = 1.99, to investigate how molecular gas, dust, and star formation relate in low-metallicity galaxies at the peak epoch of star formation. We target our sample with several deep Atacama Large Millimeter/submillimeter Array and Very Large Array datasets, and find no individual detections of CO[4–3], CO[1–0], or dust, in stark contrast to the  >60 per cent detection rate expected for solar-enrichment galaxies with these MS H α star formation rates (SFRs). We find that both low- and high-density molecular gas (traced by CO[1–0] and CO[4–3], respectively) are affected by the low enrichment, showing sample average (stacked) luminosity deficits >0.5–0.7 dex below expectations. This is particularly pertinent for the use of high-J CO emission as a proxy of instantaneous SFR. Our individual galaxy data and stacked constraints point to a strong inverse dependence ∝  Z γ of gas-to-dust ratios (G/D) and CO-to-H2 conversion factors (αCO) on metallicity at z  ∼ 2, with γG/D <−2.2 and |$\gamma _{\alpha _{\rm CO}}\lt $| −0.8, respectively. We quantify the importance of comparing G/D and αCO versus metallicity trends from the literature on a common, suitably normalized metallicity scale. When accounting for systematic offsets between different metallicity scales, our z  ∼ 2 constraints on these scaling relations are consistent with the corresponding relations for local galaxies. However, among those local relations, we favour those with a steep/double power-law dependence of G/D on metallicity. Finally, we discuss the implications of these findings for (a) gas mass measurements for sub- M * galaxies, and (b) efforts to identify the characteristic galaxy mass scale contributing most to the comoving molecular gas density at z  = 2. [ABSTRACT FROM AUTHOR]