The parallel-edge path sorting conjecture

For $r\geq1$, let $r\mathbb{Z}$ be the graph obtained from the infinite path $\mathbb{Z}$ by replacing each edge with $r$ parallel edges. Parallel-edge path sorting conjecture. For all $r\geq1$, $r\mathbb{Z}$ sorts $\Delta^n$ whenever

$$n\equiv0\pmod{2r}.$$

This is a conjectural extension of the sorting theorem for the infinite path to paths with parallel edges. The source gives no proof or resolution beyond the stated conjecture.