The generalized parallel-loop sorting conjecture

Let $S\subseteq\mathbb{Z}$, and let $r(\mathbb{Z}\langle S\rangle)$ be obtained from $\mathbb{Z}\langle S\rangle$ by replacing every edge, including every loop, by $r$ parallel copies. Suppose $n\in\mathbb{N}$ satisfies

$$\mathrm{min}(\widetilde{n\delta_0}^{\mathbb{Z}\langle S\rangle})<\mathrm{min}(\widetilde{(n-1)\delta_0}^{\mathbb{Z}\langle S\rangle})$$

and

$$\mathrm{max}(\widetilde{n\delta_0}^{\mathbb{Z}\langle S\rangle})>\mathrm{max}(\widetilde{(n-1)\delta_0}^{\mathbb{Z}\langle S\rangle}).$$

Generalized parallel-loop sorting conjecture. Then $r(\mathbb{Z}\langle S\rangle)$ sorts $\Delta^{rn}$ for each $r\geq1$. This simultaneously generalizes the looped-path and parallel-edge conjectures, but the source supplies no proof of the general assertion.