The looped-path sorting conjecture

Let $S\subseteq\mathbb{Z}$, and let $\mathbb{Z}\langle S\rangle$ be the graph obtained from the infinite path $\mathbb{Z}$ by adding one loop at each vertex in $S$. For a chip configuration $\mathcal{C}$, write $\widetilde{\mathcal{C}}^{\mathbb{Z}\langle S\rangle}$ for its stable stabilization on this graph. 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

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

Looped-path sorting conjecture. Then $\mathbb{Z}\langle S\rangle$ sorts $\Delta^n$. Here, sorting means that every labeled stabilization of the initial configuration $\Delta^n$ is in sorted order. This generalizes the main sorting theorem for the infinite path and includes the cases $S=\varnothing$ and $S=\{0\}$, while the general balanced-loop case remains open.