Sun-Yang-Zuo's twisted Higgs-de Rham flow conjecture

Let $D=0+1+\lambda+\infty$ on $\mathbb{P}^1$, and consider logarithmic graded semistable Higgs bundles on $(\mathbb{P}^1,D)$ whose underlying graded vector bundle is isomorphic to $\mathcal{O}\oplus\mathcal{O}(-1)$. Their moduli space is identified with $\mathbb{P}^1$ by the unique zero of the Higgs field. For a lifting of $(\mathbb{P}^1,D)$ to $W_2$, let $\psi_p:\mathbb{P}^1\to\mathbb{P}^1$ be the self-map induced by the twisted Higgs-de Rham flow, and let $f:E\to\mathbb{P}^1$ be the elliptic-curve double cover branched over $D$, with $f^{-1}(\infty)$ as its identity. Sun-Yang-Zuo's conjecture. For $p\neq 2$ and any lifting of $(\mathbb{P}^1,D)$ to $W_2$, the diagram

$$\begin{CD} E @>{[p]}>> E\\ @V{f}VV @VV{f}V\\ \mathbb{P}^1 @>{\psi_p}>> \mathbb{P}^1 \end{CD}$$

commutes; equivalently, $f\circ[p]=\psi_p\circ f$. This predicts that the twisted Higgs-de Rham dynamics on the moduli space is induced by multiplication by $p$ on the elliptic double cover. The claim is stated in the source as Conjecture 4.8 of Sun, Yang, and Zuo; the supplied parser gives no resolution status.