Toric promotion order conjecture for path graphs

Let $G$) be a path graph with $n$ vertices, and for $1\leq h\leq \left\lfloor n/2\right\rfloor$ let $\zeta^{(h)}=12\cdots(n-h)n(n-1)\cdots(n-h+1)$. For a permutation $\pi=\pi_1\cdots\pi_n\in S_n$, write

$$\operatorname{TPro}_\pi=\tau_{\pi_n}\tau_{\pi_{n-1}}\cdots\tau_{\pi_1}\colon\Lambda_G\to\Lambda_G.$$

Toric promotion order conjecture. If $G$ is a path graph with $n$ vertices and $1\leq h\leq \left\lfloor n/2\right\rfloor$, then the operator $\operatorname{TPro}_{\zeta^{(h)}}\colon\Lambda_G\to\Lambda_G$ has order $h(n-h)$. This generalizes the known $h=1$ case, which follows from the forest theorem mentioned in the source. The conjecture concerns the orbit structure of toric promotion operators for path graphs.