Positive traces on quantized abelian Coulomb branches

Let $A=A_{G,N}^{\mathrm{\hslash }=1}$ be a quantized Coulomb branch with an antilinear automorphism $\rho$. A map $T\text{\hspace{0.17em}⁣}:A\to \mathbb{C}$ is called a positive trace if $T(a\rho (a))>0$ for all nonzero $a\in A$. Positive traces on Coulomb branches appear in the study of supersymmetric gauge theories. We classify positive traces on all abelian Coulomb branches, meaning $G=({\mathbb{C}}^{\times }{)}^d$ is a torus.