Large statistical samples of quasar spectra have previously indicated possible cosmological variations in the fine-structure constant, $alpha$. A smaller sample of higher signal-to-noise ratio spectra, with dedicated calibration, would allow a detailed test of this evidence. Towards that end, we observed equatorial quasar HS 1549$+$1919 with three telescopes: the Very Large Telescope, Keck and, for the first time in such analyses, Subaru. By directly comparing these spectra to each other, and by `supercalibrating them using asteroid and iodine-cell tests, we detected and removed long-range distortions of the quasar spectras wavelength scales which would have caused significant systematic errors in our $alpha$ measurements. For each telescope we measure the relative deviation in $alpha$ from the current laboratory value, $Deltaalpha/alpha$, in 3 absorption systems at redshifts $z_{mathrm{abs}}=1.143$, 1.342, and 1.802. The nine measurements of $Deltaalpha/alpha$ are all consistent with zero at the 2-$sigma$ level, with 1-$sigma$ statistical (systematic) uncertainties 5.6--24 (1.8--7.0) parts per million (ppm). They are also consistent with each other at the 1-$sigma$ level, allowing us to form a combined value for each telescope and, finally, a single value for this line of sight: $Deltaalpha/alpha=-5.4 pm 3.3_{mathrm{stat}} pm 1.5_{mathrm{sys}}$ ppm, consistent with both zero and previous, large samples. We also average all Large Programme results measuring $Deltaalpha/alpha=-0.6 pm 1.9_{mathrm{stat}} pm 0.9_{mathrm{sys}}$ ppm. Our results demonstrate the robustness and reliability at the 3 ppm level afforded by supercalibration techniques and direct comparison of spectra from different telescopes.
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