Strict comparison holds in the uniform Roe algebra of a discrete amenable group

Let $\Gamma$ be a countable discrete amenable group, and let $A=l^\infty(\Gamma) \rtimes \Gamma$ or $A = \mathrm{C}(M) \rtimes \Gamma$, where $(M, \Gamma)$ is the universal minimal set of $\Gamma$. It is shown that if $a, b \in A \otimes \mathcal K$ are positive elements such that $$\mathrm{d}_\tau(a) < \mathrm{d}_\tau(b),\quad \tau \in \mathrm{T}(A),$$ then $a$ is Cuntz subequivalent to $b$.