The multiplication-by-p conjecture for the Higgs-de Rham self-map

For $bbP^1_{W(\mathbb{F}_q)}$ with marked points $bb\{0,1,\infty,\lambda\}$, let $\pi:\mathcal{C}_\lambda\tobbP^1$ be the double cover ramified at these points. Let $[p]:\mathcal{C}_\lambda\to\mathcal{C}_\lambda$ denote multiplication by $p$, and let $\varphi_{\lambda,p}:bbP^1\tobbP^1$ be the self-map induced by the Higgs-de Rham flow.

Multiplication-by-p conjecture. The diagram commutes:

\xymatrix{ \mathcal{C}_\lambda \ar[r]^{[p]} \ar[d]^{\pi} & \mathcal{C}_\lambda \ar[d]^{\pi} \\ bbP^1 \ar[r]^{\varphi_{\lambda,p}} & bbP^1 \\ }

The conjecture identifies the Higgs-de Rham self-map with the map induced by multiplication by $p$ on the associated elliptic curve. It was checked in the paper for all primes $p\leq 50$, but no general proof is given.