$K$-theoretic Coulomb branches of quiver gauge theories and cluster varieties

For $\mathrm{\Gamma }$ a quiver without 1-cycles, we show that the Braverman–Finkelberg–Najakima quantized $K$-theoretic Coulomb branch algebra ${\mathcal{A}}_{\mathrm{\Gamma }}$ of the corresponding quiver gauge theory is isomorphic to the quantized universally Laurent algebra (upper cluster $X$-algebra) associated to an explicit initial seed ${\mathrm{\Pi }}_{\mathrm{\Gamma }}$. We also show that ${\mathrm{\Pi }}_{\mathrm{\Gamma }}$ admits a cluster Donaldson–Thomas transformation as defined by Keller.